The "it has a sword/axe/hammer so it's a fighter" and "it punches and/or kicks so it's a monk" ideas need to die in a fire.This I completely agree with, especially about superheroes.
A big fire.
If it isn't a warrior strictly devoting 75% or more of its train to the pure pursuit of martial combat with the use of weapons and armor, it isn't a fighter.
Super héroes and Mythic heroes tend to powers that allow them to ignore aspects of weapons and armor combat and shift them out of the fighter class. A few of them have powers that allow them to focus on martial combat arts even more like Captain America and Black Panther.
DC cartoons tend to display this very well where martial artists there beat up the super powered because the Superpowered often don't know how to fight.