I think it depends a lot on the nature of your world. In most of our myths, monsters are a product of some breach of the ‘natural order’ for instance, I don’t believe a centaur is ever described as a monster, though they are often depicted as a menace. The Minotaur on the other hand was a punishment devised by the gods, and I think could be considered a monster.

If you don’t want this system you could simply state the monsters are an especially dangerous type of animal, and could include, say, a specific dangerous wolf as a warg while generally excluding wolves that keep to themselves.

What is and what isn’t a monster can be (probably should be) culturally defined unless you really need something to only interact with monsters, like Pokémon, or D&D’s distinction between charm person and charm monster.