Arial Black
Adventurer
It's your PC, but the DM has the final decision in the rules of the game, meta or not. Multiclassing is an optional rule and the DM chooses not to use it the player has the choice to accept that, find a different game, or volunteer to DM themselves.
A player demanding that the DM uses multiclassing is no different than demanding that the DM uses minis and a grid for tactical combat or that they play in Dark Sun when the DM is putting together a more traditional sword and sorcery game. Basically you don't have a right to it. You can discuss it with the DM and the rest of the group, but if in the end the DM chooses not to use an optional rule or even change the default rules the only choice you have as a player is play by the DMs selection or don't play.
"With great power comes great responsibility."
The DM didn't conquer the players and require them to do whatever he says. The DM rules by consent.
It is incumbent on the DM to take his job seriously, and make his decisions in a fair and open manner.
If the DM has a rule, he should be self-aware enough to avoid fooling himself about the reason he has such a rule. If he realises that his rule makes no sense or is just about what he likes or dislikes without regard to what the players may like, he should think again.
Does the rule make a difference to the DM's world or game?
A rule can make a difference to his world. For example, the DM may have created a world where arcane magic simply doesn't exist; as a consequence, no class or ability can grant arcane spells. The DM tells the players (before they create their characters!) that there are no bards, sorcerers, warlocks or wizards, and that there are no eldritch knights or arcane tricksters, and no Arcana domain, and feats cannot reproduce arcane magic.
Fine. But it would not be fine to ban Fighters or Rogues on the grounds that there is 'no such thing' as 'fighting' or 'rogue-ing(?)' in his world. That would be absurd.
A rule can make a difference to a game, in terms of how a particular game mechanic affects the game-play itself. For example, he might introduce the Flanking rule (or take it away again) from the DMG on the grounds that there should be a benefit for such positioning (or take it away because that rule gives far to great a benefit for far too little difficulty, and he doesn't like the inevitable 'Conga Line Of Death' that this rule encourages). But he cannot reasonably say that it's not allowed to stand in a flanking position! He can just say that there is no mechanical benefit for standing in that position.
One common problem with DMs is the tendancy to think, "I don't like it, therefore no-one else can have it!" "I don't like pizza, therefore no-one else is allowed to eat pizza" is just weird. "I don't like playing paladins, so if any player plays one I'll punish them by giving 'fall or fall' scenarios until they stop" is the kind of passive/aggressive bollocks that shows you're not ready to DM.
So, if you're thinking about a rule, ask yourself why you're making the rule. Is it really because it goes against your vision for your world? Is it really because it messes with game balance? Or is it just because you don't like it, without any rational reason and without regard for differences in personal taste or player agency.
Does multiclassing affect your vision for your world? No, it simply cannot; it is a meta-game construct that is not observable in the game world.
Does MCing mess with game balance? No, it is already balanced; there is no part of a MC PC that you don't already allow in your game.
It's just your personal taste.
As a quick check, think about if the opposite rule would make sense or not. The opposite of 'no multiclass PCs' is 'no single class PCs (above 1st level)'. Would that rule even make sense in the world? Would it make sense in game play?
If a player wants something for his PC and asks you if it's okay, if you say 'no' then you should have a valid reason for disallowing it.
"Can my human PC be left handed?" "No!" Doesn't make sense.
"Can my PC start at 20th level" "No, the PCs are starting at the same level, and for this campaign we are starting at 3rd". The game has to be fair.
"Can I play a cleric?" "No, Dragonlance at this time has no clerics because the gods abandoned mortals". Valid world creation.
"Can I play a fighter?" "No, my world doesn't have fighters because there is no such thing as....fighting?" But barbarians can still hit each other?
"Can I be muticlass?" "No!" "Why?" "Because in my world if you learn how to fight then you cannot sell your soul, or if you worship a god then you cannot be as skillful as someone who doesn't." WTF?
"Now that we've leveled up, can my Rog 3 take a level of fighter?" "No, because I don't allow fighting styles or healing". "Yes you do, Jeff is playing a Ftr 4; how can you say that you don't allow Ftr 1? How can you say that rogues cannot learn how to fight when they already fight by stabbing people?"
There is no valid reason to ban MC. It's just personal taste. DMs should recognise this and not impose their personal tastes on the players. "I like anchovies on my pizza, therefore everyone else must have anchovies on their pizza too!"