There are a few reasons for a spellcaster to be in melee (most of which you mentioned.) I'll cover them one by one.
One is auras that buff nearby allies. The Paladin wins this one. Honestly, no other class even comes close. However, outside that, the Paladin's primary focus is on buffing his own melee attacks with the [X]-ing Smite spells, which isn't exactly what you're after. Also, as a half-caster, the Paladin is almost certainly not what you're after.
The next is spells that damage nearby enemies. The standout here is the Cleric's Spirit Guardians, which, once cast, damages everything around you for up to 10 minutes without requiring any further actions on your part. Other notable spells include the Paladin's Destructive Wave and the Warlock's Arms of Hadar. There's not too many spells in this category, unfortunately. The Elemental Player's Companion addded some good new ones to look for - Thunderclap is a cantrip that's just perfect for a melee spellcaster, and Earth Tremor is a pretty awesome too.
Reason number three is if you want to be attacked for some reason. The Wizard's Fire Shield is probably the best spell for this, but the Warlock's (or Tiefling's) Hellish Rebuke or the Tempest Cleric's Wrath of the Storm ability work here too.
Next up, spells that have their origin on you. Burning Hands, Cone of Cold, Thunderwave, etc. There's a lot of these, but none of them really stand out, no class is particularly better at them, and most of them don't really require you to be in melee anyway. They're good, and you'll want to have some of them, there's just not much to say about them here.
Finally, melee range attack spells. (And yes, I saved the best till last.) There is a definite winner here: Vampiric Touch. A minute of powerful melee spell attacks that do a fair bit of damage and heal you? Perfect. It's available to a few classes, but the one that gets the most out of it is a Necromancy Wizard, because Grim Harvest will get you even more healing when you kill something with Vampiric Touch. If your DM will allow it, the Death Cleric from the DMG also gets pretty good with this spell, with Touch of Death, Inescapable Destruction, and eventually, Improved Reaper (!!!).
I'd probably recommend the Necromancy Wizard or the Death Cleric, but whatever you choose, you are going to run into one problem: concentration. Most of the good spells I've mentioned require concentration, which means you can only use 1 at a time, and you are liable to lose the spell when you get hit by attacks. Which, as a melee spellcaster, is probably going to be often. Concentration checks aren't particularly difficult, but if you have to make a lot of them you might consider investing in a method of gaining advantage or proficiency on them. The War Caster feat gives advantage, while the Resilient feat, Sorcerer, or Barbarian give proficiency. Of course, there are a lot of ways to give advantage on or boost any roll, so you might not find this a problem.
On Wizard schools specifically:
Abjuration has some great benefits, but the only one that is particularly good for a melee is the Arcane Ward, which effectively soaks one hit per day, plus a little bit more every time you cast an Abjuration spell. Not bad, but not great.
Conjuration has Focused Conjuration, which allows you to ignore damage for concentrating (great!) but only for Conjuration spells, of which there aren't many that suit you.
Divination has nothing that works specifically for your purposes.
Enchantment actually has one ability that's great for this: Hypnotic Gaze. It effectively allows you to lock down a single creature in melee.
Evocation is great for evocations (as you might expect), and most of the damage spells with you as the origin are evocation, so definitely worth a thought.
Illusion has an auto-miss reaction, which is cool, but not much else.
Necromancy has the aforementioned Grim Harvest, which is pretty good, and Inured to Undeath, also pretty good (but only when fighting undead), but its other features are more geared around having an army of undead fighting for you, which is pretty powerful, but not really what you're after.
Transmutation doesn't have all that much for a melee, except that you can get a free proficiency in Constitution saving throws, which will help you maintain concentration.
On Shield Master specifically: for you, it's almost completely junk. It boosts your Dexterity saving throws, and nothing else. I'd really recommend against taking it. Heavily Armoured, on the other hand, will take your AC from a maximum of 17 (assuming you have a Dex modifier of +2), to 18, with no investment in Dex required. With the houserule in place that gives you shield proficiency as well, it pushes your AC up to 20 - at this point it goes from good to downright amazing. AC is quite rare and valuable in 5e, especially if you're used to 3.5 and everyone walking around with AC 30.
Talk it through
This turns out to be the answer for a lot of issues like this. D&D is a social game and the only way to resolve many of the issues that come up between players is by talking about it. In an MMO, it's easy to just drop the group, kick a player, or whatever...a D&D party has a lot more investment in it.
The best solution to this sort of problem is to sit down with the other player and talk it over. If you aren't comfortable with this, go talk to your DM. They should be able to help you. Here are a few pointers...
Talk in private
You don't want to call the other player out in front of the rest of the group, that's a quick way to make someone defensive. Instead, catch them before or after the game when you can talk privately.
Focus on the game, not the person
You don't want to rip into another player over this...again, that just makes them defensive. Instead, focus on the game and the characters, and how their abilities can interact. Talk about your respective roles and responsibilities, and how you can best work together to enjoy the game.
If they are more experienced in the system (or at RPGs in general) than you, consider approaching it as asking for advice
To this particular case...
"Hey, I noticed that our character's abilities are kind of interfering with each other. Your Sorcerer keeps knocking enemies all over the place, and I'm trying to round up enemies to keep them focused on my Paladin, so they don't go kill the Druid. I mean, the opportunity attacks I'm getting are nice, but I'm going to be a lot more effective if I can keep enemies close to me. Do you have any suggestions for how we can fix it so our characters compliment each other more?"
Talk to your DM
This one doesn't apply as much in your particular case...but for broader applicability...consider a sneaky character in a party full of rushdown players ("who needs scouts, let's just charge in!"). This is a case where the entire game is clashing with a particular character's skillset, rather than a single character messing up another's ability to operate.
In this case, the best option is to talk to the DM about helping develop situations where your character can shine. Again, the goal here is to have fun...most experienced DMs will look at their playerbase and try to engineer situations where all of them get their chance in the spotlight...but sometimes a DM needs a nudge.
Solve problems with out-of-game solutions whenever possible
One thing you absolutely do NOT want to do is start having your character act out against their character. In an MMO, it's pretty common for a Tank who gets mad at another character to stop protecting them. "Whoops, rogue pulled again, I'll taunt it after he's dead."
Don't do this. All it results in is mad players, hurt feelings, and arguments that can tear the entire game apart. Especially don't do this if you have not talked to the other player about what's going on. They (probably) don't know they are upsetting you. They might even think they are being helpful or making you more effective.
Talk to people to solve problems...don't take it out on them in-game.
Regarding this Particular issue...
You referred to World of Warcraft several times, so I'll reference that game as well. Essentially what's happening here is that you have a character who keeps pulling mobs off the tank. Imagine a character (like an Elemental Shaman) using a knockback ability to punt mobs off the Main Tank. Sure, it does damage and momentarily halts that mob's ability to do damage, but they are hampering the abilities of the tank and it's probably going to get the Healer killed. In a typical WoW group, such a player would get kicked from the party, or the tank and healer would drop out of the PuG. In D&D, kicking a player or dropping out is a much bigger deal, so the better option is to work through the problem in a social manner.
Talk about it, figure out what's happening, and how you can all work together to fix it. After all, the ultimate goal of D&D is to have fun!
This particular case is more of an issue of the Controller not realizing the best way that his powers can be used alongside yours than him having a 'different playstyle.' Sure, knocking enemies away from you gives you an Opportunity Attack. But if he instead used his 'move them around' powers to pile enemies up on top of your Defender, you can use all of your awesome defensive powers to wreak absolute havoc on your mutual foes.
In 4E, at least, the optimal use of a Controller is to lock down a few enemies to keep them out of the fight so you can deal with them later and/or help keep the enemies piled on the Defender. If an enemy wanders off to go attack someone (like the Leader), then the Controller is best served to knock that enemy back over to the Defender or lock them down so they can't hurt anyone.
If the Defender has most of the enemies rounded up, the Controller does best taking a few of those enemies, and tying them up somewhere away from the rest of the party so they aren't dealing any damage to anyone. The same way that, in a WoW Raid, you may CC a few of the enemies to keep the pressure off the tank, so the tank doesn't have to fight quite as many enemies at once.
Best Answer
Generally-speaking, AC is stronger the more AC you have.
For instance, an enemy that has to roll a 19 to hit has a 1/10 chance to hit, and would have a 1/20 chance if they needed a 20. A single +1 AC can effectively double your defense (assuming it's high enough).
You'll probably get more value adding AC to those that anticipate being attacked, while your other members should get the non-AC benefits (THP, save bonuses, etc).
Of course, if your DM is the kind to ignore the 20 AC Fighter, it's probably better to spread out those items or have the Fighter pick up things that make him a better target (like the Sentinel feat or the Cavalier subclass).