Your AC will be the better of the two (creature's or unarmored).
PHB pg. 67:
Your game statistics are replaced by the statistics of the beast, but you retain your alignment, personality, and Intelligence, Wisdom, and Charisma scores. You also retail all of your skill and saving throw proficiencies, in addition to gaining those of the creature.
Although this looks like you lose unarmored defense, it already adds exceptions. The last bullet of the description of Wild Shape states:
You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so.
This means that you can choose Unarmored Defense, meaning you can choose to use the better of the two Armor Classes.
There are no requirements about multiclass stats for anything other than multiclassing. The best reasoning as to why this is true is the fact that the multiclass rules state that the necessity for high attributes for multiclassing is because you are learning a new set of abilities quickly. You are expected to have learned your original class over the course of years, where multiclassing can, in many games, happen overnight.
Important note:
When you wild shape your physical stats do change, so remember that your AC from the monk's unarmored defense will adjust based on your new Dexterity modifier in the creature's form. This also means that a Barbarian/Druid would calculate their unarmored defense AC from "10+Dex Mod+Con Mod" based on their new stats in beast form. Additionally, a wild-shaped druid that has a buff increasing their Dexterity will increase their AC according to the increased Dex mod value.
http://dnd.wizards.com/articles/features/rules-answers-january-2016
Natural Armor: 10 + your Dexterity modifier + your natural armor bonus. This is a calculation method typically used only by monsters and NPCs, although it is also relevant to a druid or another character who assumes a form that has natural armor.
Additionally
It is also important to note that your mental stats (Int, Wis, Cha) remain the same. This means that you use your Wisdom modifier and the animal form's Dex modifier to calculate the monk's version of Unarmored Defense for a monk/druid.
This does not apply to the standard armor class calculations for the animal's natural armor calculation, only the Unarmored Defense armor class calculation.
So, for a Druid/Monk/Barbarian, you would have three possible armor class calculations available:
- 1) Natural Armor = 10 + (beast form's) Dex mod + (beast form's) natural armor.
- 2) Monk UD = 10 + (beast form's) Dex mod + (character's) Wis mod.
- 3) Barb UD = 10 + (beast form's) Dex mod + (beast form's) Con mod.
You would select the best of the three Armor Classes as your AC in each beast form.
Problem?
In this particular case, nothing. Any concern you might have about "overpowered" is mitigated by his stats being at unimpressive levels. From your previous question you indicate that the players are not leaning toward min-max, power gaming, nor optimization in a munchkinny way. He's using finesse weapons that in time may limit highest damage potential: but if they aren't in min max mode, that does not seem to matter to the group.
At higher levels, he may get a little frustrated with melee damage since there are no two-handed finesse weapons and there are fewer choices in weapons ... but a lot of that gets into what fighting styles he uses and chooses. (And you control what magical weapons, if any, become available so you can tailor that to this party/team. Our DM did just that for our Bard, who is a rapier using singer).
At higher levels he'll benefit from those Divine Smite nova shots. When paired with sneak attacks from his rogue levels, he should be able to keep his damage in a "good enough" zone if the party fights well as a team.
The other place a low Strength would be a Paladin issue is in armor selection: if he wants Chain, Splint or Plate, he needs 13 / 15 / 15 strength respectively. With the proposed Dex build and Rogue multiclass, he's unlikely to want them since they give Disadvantage on Stealth checks. He can wear Breastplate with no minimum Strength requirement -- allows up to +2 of his Dex AC bonus, and no disadvantage for stealth. At 400 GP it may be a level or two before he can afford it. A 50 GP chain shirt may be his best for the time being.
From @Ethan's comment: With leather armor, a shield, and 20 Dex (After a ASI's) he will end up with 19 AC, 1 lower than full plate+shield, but the Piradin will be good at all +Dex skills. This makes for solid function when full plate isn't a good idea -- such as waterborne adventures. His damage output will be identical to a strength based shield using paladin (barring difference in MC class features).
If this is a role and story based move, all the more reason to go with it. It sounds like fun - which is why we play. As long as the player knows where the difficulties are up front with the stats as is, the player is taking on a challenge. Good on 'em!
I'd recommend a ruling to let him proceed as desired. (And if later on they finds some gauntlets of ogre strength ... )
Best Answer
The multi-classing rules say on PHB page 163, under “Prerequisites”:
Emphasis mine. You do in fact have to meet the prerequisites for BOTH classes if you want to multiclass, whoever told you that is correct.
It's important to note that stat boosts from items that aren't permanent, such as Ogre Gauntlets, do not count as meeting the stat requirements. (Thanks to tardigrade for pointing that out.)