Do you even talk to the GM about it? Or you just reskin away and call him preposterous?
Or the GM doesn't care what you do?
Do you play regularly & if so how does this work?
When I actually play (as opposed to the
theory of play) I find that the more you and the DM bounce ideas off each other, the more each of you like the character. Each of you will think of things that the other wouldn't, inspired by ideas
they have. The whole is greater than the sum of its parts. In this scenario, the arguments we've been having on this thread have been solved before they've arisen.
Ideally, the DM will be open to ideas for the world and for his own PC that have been put forward by a player, but the DM has the final say. I'm not saying that his rulings
must make sense, but they
should.
Ideally, the player will be open to character ideas from the DM, but
within the rules permitted by the DM, the player has the final say on the characterisation and concept of his own PC. If he doesn't, it's just 'DM story time', with the players as mere listeners.
A bit extreme, perhaps, but if the player doesn't have agency of his own PC he might as well stay at home and read a book.
Just as for the DM, the player's ideas don't
have to make sense (although they should); the player just cannot break the
rules. Fluff is not rules.
But the risk players take by 'not making sense' is that the DM doesn't allow that PC. The player has an incentive to make a PC that fits into the world. The DM has an incentive to make reasonable rulings or he loses players.
So there is a happy medium. Lines of demarcation say that the player controls
and creates his own PC (according to the rules), while the DM controls
everything else in the world.
It's a judgement call. If we are playing a Lord of the Rings game using 5E rules , then if I want to play a Space Ranger with a rocket pack and the DM says no, is the DM being reasonable? I think we'd agree that he is. But if the player wants to play a PC with some ranger class levels without belonging to The Rangers of the North, and has a backstory that explains how he learned his weaponskills, favoured enemies, spells, every ability of the class, is the DM unreasonable if he says no? I think so, but I wouldn't be surprised if he said no.
But if the DM allows the monk class and allows multiclassing, if the DM says that it's
impossible to learn supernatural combat techniques in any other kind of building than a monastery, he's overstepped the mark.
The monk class was inspired by Shaolin monks, just as paladins were inspired by the Twelve Peers. But Shaolin monks' martial arts was created accidentally IRL when the head monk thought that his monks were unfit and devised a way to get them fit. Over the centuries people created variants, left and created new monasteries, or left and trained students privately. Some of those students also trained others privately, without the new guy ever seeing a monastery.
Once the basic idea of this kind of unarmed combat had been thought of (and it's only an accident of fate that they were created in a monastery at all), the idea exists within the world and masters can train students wherever they like.
In a world where magic exists and magic can be harnessed as 'ki', then it doesn't matter if 'ki' has different names in different places (just like in our world), doesn't matter if the training takes place in a monastery. The DM has ruled that these techniques (this class) exists in his world, and it's unreasonable, even illogical, to say that no-one in 10,000 years could possibly have trained someone outside of a monastery.
Once anyone ever trained a student privately, then they could start a line of non-monastery trained 'monks'. There is no class ability from the monk class that requires religion or a type of building without which those techniques are impossible.
The DM could very well concepualise a class ability like, say, Deflect Missiles and say that The Dark Disciples of the Red Redemption call this technique 'Cobra Striking Hummingbird', and he can certainly say that Cobra Striking Hummingbird is only taught by The Dark Disciples. But the player can certainly say that the Lachrymae Shevarash call Deflect Missiles 'Falling Rain' technique, and this training is (supposedly) only taught to other Tears of Shevarash.
Now, to an observer, each technique may look very different, but they share the same game mechanic: Deflect Missile. But it is unreasonable to say that Cobra Striking Hummingbird is allowed in your world, but Falling Rain technique cannot
possibly have been learned outside of a monastery!