In short, yes.
There are a few things that make monk on par with the other melee fighting classes. I'll list a few of them here:
Dex to attack and damage
Monks can apply dex to attacks and damage automatically at level one.
Extra attacks
Monks are the only class that gets 2 attacks per round at level 1, and they keep up with Fighter attacks per round all the way to 20.
Class features apply to monk weapons
What counts as a monk weapon has been expanded a lot, and most of the monk class features (including increased damage) apply to monk weapons and unarmed strikes equally. In addition, the slew of exotic weapons that monks used to get have been condensed into reflavours of simple weapons. For example, nunchaku are now mechanically clubs, rather than a separate weapon type.
Less MAD
Monks in 5e have much less of a Multiple Attribute Dependency problem. Since monks get Dex to attack and damage for free, Strength is largely unnecessary. Int and Cha are as useful as they were in 3e. There are a few things that Int and Cha are useful for in both systems, but they're hardly essential.
Wisdom affects the same sort of stuff in 5e as it did in 3e, but it's less important due to other mechanical changes. Since "flat-footed" doesn't cause you to lose Dex Ac anymore, using Dex to increase AC is just as good as increasing Wis. You can probably end up with similar AC as a fighter by then end game.
Monks need to care about 3 stats, but they have basically the same number of dependencies as a fighter or barbarian. The fighter and barb both need Str and Con, with a secondary focus on Dex, and a monk needs Dex and Wis, with a secondary focus on Con. Since the stat cap is 20, it's actually not difficult to have a 20 and a couple 16-18s, as well as a 12 or two.
Better class features
The monk class features in 5e are similar to the ones in 3e, but they're a bit better, and a bit more reliable than they were before. I haven't done any real analysis on this, this is really just based on a first glance.
Interesting paths
There are three monk paths in the PHB, all of which do cool things. There's one for a classic monk that's similar to the one in 3e, but the other two add some really interesting things to the class that broaden the kind of situations that the monk can handle, which is a problem that mundane classes have had since the dawn of time. :)
Overview
Broadly speaking, the monk does less damage than the Barbarian, and has less tanking ability than the Fighter, but is a very strong and versatile mobile melee fighter. A better character optimizer might have a different opinion, but as a player who doesn't make broken characters, the monk looks like it's on par with the Barbarian and Fighter.
There're at least two ways...
The following let the barbarian actually rage using his own kind of rage more often than normal. "Whirling frenzy is otherwise identical to the standard barbarian rage" (UA 66), so these should work with that, too.
The 2nd-level Drd spell blood frenzy [trans] (Spell Compendium 33) causes 1 touched creature to rage as if it used its special ability rage normally "but this rage does not count toward the creature’s number of rage attempts per day." Note: Rage attempts?
This is likely the best choice, equipping an ally with a wand of blood frenzy (2nd-level spell at caster level 3) (90 gp/charge) and the barbarian himself with a potion of blood frenzy (2nd-level spell at caster level 3) (300 gp; 0 lbs.) in a potion bracer (Sharn: City of Towers 158, 159) (100 gp; 1.5 lbs.) for emergencies (that is, if the DM's not ready for the ready-drink helm to enter her campaign.)
The shoulders slot item mantle of rage ("New Magic Items" 2) (6,000 gp; 1 lb.) grants the wearer, if a barbarian, the ability to rage 1 additional time per day, and, if not a barbarian, the ability 1/day to rage like a level 1 barbarian.
...And a clutch of edge cases
In addition to the 3rd-level Sor/Wiz spell rage [ench] (PH 268) et al. (which isn't really rage but a buff spell with attached penalties), there are a few lesser alternative sources of barbarian rage not involving taking levels or feats.
- The gutworm (Fiend Folio 220) is a symbiont that at will can cause the host to "enter a frenzied state identical to a barbarian’s rage" for 4 rounds, but this rage explictly provides bonuses like the typical level 1 barbarian's rage. Note: More rage, but not the right kind of rage.
- The face slot item mask of fury (Dragon #324 76) (12,000 gp; 2 lbs.) 2/day grants the wearer the ability to take either a free action (if he's been wounded within the last hour) or a standard action and deal himself 3 hp of damage (if he hasn't been wounded within the last hour) to enter a rage "as the barbarian ability." Note: Whether this can be a barbarian wearer's own rage or must be the rage of a level 1 barbarian is the DM's call.
- The supernatural ability past life (DMG2 159-60) has one option that 1/day can grant the possessor the ability to enter explicitly a greater rage as a level 11 barbarian. This special ability is nominally for NPCs and has a (ridiculous) Level Adjustment of +5. Note: Probably too late for this barbarian and too stupid for any PC (or, for that matter, any PC's follower or cohort).
Yes, that last scrapes the barrel's bottom.
Best Answer
Unfortunately, you have selected a massively disparate subset of the classes. Fighter, monk, ninja, and rogue are four of the weakest classes published for all of 3.5 (with Complete Adventurer’s ninja actually probably the worst here), while cleric, shugenja, sorcerer, and wizard are four of the strongest classes published (especially the cleric and wizard).
And ultimately, those spellcasters are so powerful that no one without spellcasting is going to be able to keep up if they’re played at anything like their potential.
If you are going to be selecting a very-limited subset of the classes, I suggest that you take the opportunity to choose better designed classes which have tighter overall balance. Your warriors are not powerful enough, your spellcasters are too powerful.
You can ignore the advice for everyone but the fighter and the monk – I still do strongly recommend the swordsage and warblade there – but even those improved options won’t be able to keep up with those spellcasters, at least not unless the people playing them are underplaying their power quite a lot. Therefore, I’d like to offer a more “complete” recommendation for every class on your list.
In all cases, I have carefully chosen the replacement class so that it can be a “drop-in” replacement – you don’t have to make up a new place in your setting for warblades, just use the warblade’s stats and rules and call it “fighter.”
Stronger warriors
The more “direct” answer to the question, there definitely are superior options for playing a “fighter” or “monk” than the fighter class or monk class. Note, however, that even these are still a far cry from the spellcasters you have selected (and, indeed, still are not quite up to the ability of even the weaker spellcasters I recommend below).
Instead of fighter, consider the warblade
The warblade from Tome of Battle: The Book of Nine Swords is an intelligent master-of-arms, a savvy tactician and a well-honed veteran of battle. His ability to master weapons is without parallel, and in battle he is relentless and iconic, rallying his allies around him.
Instead of monk or ninja, consider the swordsage
Also from Tome of Battle, the swordsage could be monk or ninja, depending on the particular choices that you make. A “monk-like” swordsage would probably focus more on Diamond Mind and Setting Sun maneuvers, while a “ninja-like” swordsage would likely focus on Shadow Hand and Tiger Claw – but either would be remiss to ignore those other disciplines.
Instead of rogue, consider the factotum
Dungeonscape’s factotum is basically “the class Indiana Jones would take if he were in D&D.” Factota do a little bit of everything, and do it well. They use their wits and cunning to make it up as they go along, and are the best skill-monkeys in the game, bar none. Awesome, super-fun class.
Weaker Spellcasters
I wrote an answer dedicated to this question here, but because that question was overly-broad, my answer was overly-long.
But the idea is still sound: the over-powered spellcasters you have chosen could be replaced quite well by other spellcasters that fulfill similar roles and have similar narrative backgrounds. Note that these spellcasters are still very powerful, however; using these is insufficient to make the fighter, monk, or ninja classes play on even footing. Even the superior classes I recommend above are at something of a disadvantage compared to most of these; the gap is just much narrower.
For this game, I strongly recomment the following replacements:
Instead of cleric, consider the divine bard
Bards are viable warriors, while not being excellent at it. The cleric would make a far better warrior than any of your warrior classes. The bard has some solid healing spells, especially when divine, and their music makes them fantastic leaders-of-men.
Alternatively, the incarnate might be excellent
The incarnate from Magic of Incarnum is an excellent replacement, flavor-wise, for a cleric, with a strong religious theme (an evil-aligned incarnate is supposed to literally be evil incarnate). Their powers lends themselves a bit more to the skill-monkey/support role than the warrior/healer role, but they’re overall an excellent class and judiciously awarding wands of cleric spells can make up a huge part of the difference.
If you specifically wanted the warrior-priest, crusader is for you
I generally recommend replacing cleric with classes that are less devastating at armed combat, because I tend to feel that the cleric’s prowess there makes it difficult to justify the existence of, e.g., paladins. But since you have not selected paladins in the first place, you might be looking for a more martial bent from your divine class. If so, the crusader from Tome of Battle: Book of Nine Swords is perfect – he heals his allies as he smites his enemies, and as a rallying point and commander of battle, he knows few peers.
Also, crusader is one of my favorite classes in the game, and my #1 recommendation to new players.
Instead of shugenja, consider the ... actually, shugenja’s probably OK
Shugenja is the weakest of the spellcasting classes that you have listed, and while it is more powerful than the other classes I have mentioned here, it’s probably close enough so as not to cause problems.
Instead of sorcerer, consider the dragonfire adept
Sorcerers, in 3.5, are thought to base their magic powers on trace draconic ancestry. The dragonfire adept from Dragon Magic, on the other hand, obviously gets their power from dragons. They gain a breath weapon and a number of invocations, which are at-will spell-like abilities. Similar style to a sorcerer, at a lower power level. Like the warlock, which I recommend as a replacement for wizard, below, dragonfire adepts tend to be a little on the low side, in terms of invocations known. Allowing them to learn a new invocation every level except those in which they gain a new breath effect is a simple solution that I have found works well.
Instead of wizard...
Instead of enchanter or illusionist, consider the beguiler
Beguilers, from Player’s Handbook II, get a small, specific spell list, tailored towards the casting of enchantments and illusions. They do get some opportunities to add to their list, but few enough to keep them under control. They’re also a lot of fun.
The beguiler might be the most powerful class here (I don't have enough experience with shugenjas to say for sure), so you may want to reduce it somewhat. Their armored spellcasting is a minor boon they can easily afford to lose, and as an Int-based class, 6+Int skill points is quite a bit – 4+Int might be more appropriate. Removing some or all of glitterdust, haste, solid fog, and/or time stop from their spell list might be good choices.
Instead of necromancer, consider the dread necromancer
Heroes of Horror’s dread necromancer is actually better than the wizard-necromancer at raising undead minions, but the dread necromancer spell list is tightly limited, like the beguiler’s, to enforce the focus on necromancy. Again, smaller list with carefully-controlled opportunities to expand it do a lot to limit the power of the wizard.
As will become a theme, changing the dread necromancer’s spellcasting to Intelligence rather than Charisma will make them more “wizardly.” The split between Intelligence for spells and Charisma for Rebuke Undead is also probably a good thing, and will slightly (but noticeably) play up the more “wizardly” necromancy of curses and debuffs, rather than the minion-master undead-army-raising that you get with Rebuke Undead, which I consider a good thing.
I’d also consider losing the ability to cast in light armor. It’s not especially powerful (it is easy to avoid Arcane Spell Failure), but the dread necromancer is on the higher-end of classes presented here, so chipping away at that is desirable. Plus, I recommended removing it from the beguiler (enchanter) and warlock (generalist wizard), leaving the dragonfire adept (sorcerer) and warmage (evoker) the only ones who have it: that seems appropriate.
Instead of evoker, consider the warmage
The warmage from Complete Arcane is the evocation equivalent of the beguiler or dread necromancer. Unfortunately, being blasting-only does leave the warmage a little flat, but they are still capable enough for keeping up with my other suggestions.
Since you want the warmage to be replacing an evoker-wizard, I would switch the spellcasting to Intelligence. This has the added bonus of making the Warmage Edge feature more useful (usually, outside of very-low levels, the bonus it provides is too small to justify investing much in Intelligence, and it just ends up being a trap, tricking players into putting too much into Intelligence).
Instead of wizard, consider the warlock
Another entry from Complete Arcane, the warlock is similar to the dragonfire adept, but much more strongly arcane, with abilities to conjure, dispel, divine, and transmute, not to mention their iconic eldritch blast. These make decent generalists; a bit on the weak side, so consider increasing their number of invocations known. I find having 1/level goes a long way. Also, eldritch blast goes up 1d6 every odd level... until 13th level, where it goes up every three levels. No one knows why it slows down like that, but it’s pretty common to just ignore that and use 1d6 per odd level, and that works well.
Making the warlock intelligence-based would also be a nice improvement of his wizardly-ness. It also helps the skill situation, though even with an Intelligence focus I’d still give the poor guy 4+Int skills. Losing the ability to invoke in light armor may be appropriate, especially if this justifies some more invocations known (say, 1/level, plus an additional 1 each time you reach a new tier of invocation? That would be awesome).