Tales and Chronicles
Jewel of the North, formerly know as vincegetorix
Its not the martial art part that makes it a little out of place in a pseudo-Europe with magic, its the built-in stuff like Astral Projection, Running on water, Jump-flight etcTo anyone who has issues with the setting for monks...
Boxing existed as one of the first martial arts from over 3000 BCE and is typical in MMA today.
Pankration was created over 4000 years ago. It is attributed to Theseus and Greek in nature. Greek mythology stressed superhuman an supernatural abilities in it's heroes.
The roots of parkour stem from a martial art developed prior to WWII based on the observations of a French naval officer observing African tribes and his position was strengthening the body, the energy, and the morality of the practitioners. The origins of parkour (African and European) fit the monk extremely well.
The class was based on a fictional Korean martial art using Remo Williams from The Destroyer so there was clearly some pop-culture reference but that doesn't mean the class doesn't apply to a typical European setting, especially in a fictional world anyway. I would point to Western martial arts and parkour as the basis here.
Martial arts are pretty universals, but martial artist that run up walls, fly in the air and project themselves in the spirit world is pretty much inspired by one single trope: the kung-fu stuff. Not even the anime characters from modern medias, which could be a good inspiration for WotC as to what people want of an unarmored martial artist, including weapon masters.