It depends strongly on the scenario. Often monsters are fauna with a supernatural element, or otherwise not very well-known or widespread. Even in historical fiction settings, say Moby Dick, something like an albino whale might be considered a monster simply because it's a dangerous animal that's also unique in some way.

The culture of the setting plays into this as well. Animals to whom superstition has been attached will be considered monsters no matter how 'mundane' they are, such as, say, the Aye-aye lemur in Madagascar. It's a funny-lookin' little goblin, but despite posing no threat to anyone, it's frequently killed when spotted because it's considered a harbinger of evil.

I think generally if the distinction is actually just between animals that exist in our world and animals that don't, that'll be either a deliberate move by the author to establish the nature of the 'monster', an accident where the author didn't consider the perspective of the setting's native characters, or, most interestingly, an implication that something has changed to introduce the 'monster'- Something about these non-Earth animals is out of place in the setting.