I think (?) you meant monstrosity and not monster???
I am using these terms as synonyms, where "monster" is natural English and "Monstrosity" is technical jargon. Maybe like saying "human" and "Homo sapiens sapiens".
That is actually more an aberration I think.
Aberration is actually similar. But D&D distinguishes between their origins.
Monsters are typically native to the material plane.
Aberrations are from the farrealms, which in my view, is an astral dominion, native to the astral plane.
Where the material plane is more about encounters with the physical, sensorial, world, the astral plane is more about concepts, ideas, paradigms.
Personally I associate Aberrations with the aggressive, destructive, behavior by the collective unconscious. It is like, if ones personal identity isnt working well, dreams and so on can often suggest alternative ways of doing things. These new ways of being are often perceived as a threat to ones identity, and can be self-destructive, but with caution can also be useful. A similar phenomena can happen at the identity of an entire culture, across many individuals.
I associate Aberration with Neutral Evil − merciless destruction. But not every Aberration is Evil.
FWIW, while people might think the whole owlbear thing is cool, it is a terrifying nightmarish creature, not a cute and fuzzy bunny.
Cultures tend to domesticate scary things. Once upon a time, djinn were terrifying. Once upon a time, elves and fairies were terrifying. Once upon a time, vampires were terrifying. Once upon a time, devils were terrifying.
Regarding the owlbear, there doesnt seem to be any characteristic that isnt a normal natural animal. Hypothetically, a mammal could evolve a beak.