Monks don't have to stick out, every culture can conceivably have martial arts. People just like the eastern theme.
It's a bit harder to justify kicking monsters but not much more than using a sword or axe. There no reason to ever willingly get in close range fights with stuff that can survive a stab and proceed to crush your head like a grape, it's just because it looks cool.
I'm going to tell you a secret...
Nearly all (excluded are things like Capoeira) unarmed martial arts are descend from the greek martial arts: Pale (Wrestling), Pygme (Boxing) und the greek MMA aka Pankration (Pan = "All", Kratos = "Power" ). It was spread from the Polis-states through Alexander's conquest of the Middle-East to the subcontinent of India and from there to the east asia (Thailand and China).
False. If anything, Mongolian wrestling is the origin of the grappling forms in East Asia. And Western MA all come from older Indo-European roots, of which both Greek and Indian subcontinent forms are branches - but so are Germanic, Nordic and Celtic.
There's nothing particularly culture-specific about punching, kicking and rasslin' anyway, they're just things humans do, with more or less codified or more or less sportified formats in various locales, with odd tricks here and there that are localized. Also a lot of changes over time depend simply on changes in rules of engagement, changes in sporting rules and how bloodthirsty vs. how squeamish spectators are (as I'm sure you know, Boxing became less bloody but
much more dangerous to the brain with the introduction of gloves, but also stances and punching techniques shifted - how different would the rules and techniques be with a "cestus," the complete opposite of the boxing glove?)
What's possibly unique to East Asia is the "internal" MA - though even there, there are hints of an equivalent to "qi/ki" (East Asia) with "prana" in India, and possibly even "pneuma" in Europe. At any rate, the common thread there is breathing exercises that condition the body's fascia, plus a very different (from normal) method of moving the body and delivering force that repurposes some of the autonomic systems in the body that balance against gravity. That would be the thing that could make the East Asian monk archetype special, but it's so recondite and obscure that nobody really knows much about it, least of all makers of RPG game rules, tv shows, or even East Asian kungfu movies