I started what I hoped would be a fun thread where we could discuss different orders or even types of monks for D&D. I regret starting this thread.
I feel your pain.
To address the OP (late to the thread...). I've had different forms of monks in different editions, mixing oriental and occidental influnences sometimes. For 5E however, since that is the current "flavor of the month" I'll give a couple examples.
First is from an "all-monk" campaign I played in online near the end of the pandemic. The DM flavored any "animal-head" races as coming from a Zootopia-like contenient far away. We had an aarakoca, leonin, and tabaxi to begin with. Each with a different subclass as well. The concept was we were emissaries from those lands seeking knowledge, trade, and perhaps even alliances with the kingdoms of the mainland. Our martial arts were derived heavily from our animal-connection. I guess you could sort of think of it like
Kung-Fu Panda but without quite as much oriental flavor?
I've also used monks as adversaries, based on the Master of the Desert Nomads module. These were more "cloistered" than kung-fu type monks.
I have one player who has monks (along with paladins) as their favourite class(es) and they play them a lot. We developed a more cloistered occidental monk (a "friar") which used ki not for physical prowess, but mental prowess in knowledge and lore. It was a fun experiment.
Finally, as others have mentioned, practitioners of martial arts come from all sorts of cultures. I was glad when the added Unarmed Fighting Style, for example.
To my mind monks are too often portrayed as the wuxia style, like bards are protrayed as minstrels. My favourite bard I played was a dwarven orator. He never had an instrument, but was more a motivational speaker-type.
Anyway, monastic or not, oriental or occidental or otherwise, feel free to break the mold with monks and give them a twist you like. I understand sometimes when a player brings in a PC who is "out of touch" with the theme or genre of the campaign, but if you failed to stipulate that prior to the player joining, just try to roll with it IMO. If you told them ahead of time, (personally) I would stand my ground and tell them "no, make something else--it doesn't fit with this campaign" and not loose a wink of sleep doing so.