You need to follow up the link to friar. Just saying…
A monastery is a place where some monks live. But many monks do not live in monasteries. Friars are one example. And some of the people who live in monasteries are not monks. And, of course, an overwhelming majority of monks do not practice any sort of martial arts.
Meanwhile, the vast majority of fictional martial arts heroes are not monks (although some may have been trained by monks). Batman, Black Widow, the Third Doctor, even Caine, the original D&D monk, are not actually current members of a religious order.
If you want to know what a monk is in D&D, read the class description. It’s quite clear that there is no requirement for them to be part of a religious order or have ever lived in a monastery. In D&D, the word “monk” means “martial arts hero”.