Owlbears have always been beasts in any campaign world I have a say in, because that is simply what they should be.
I feel like anyone who actually wants to include owlbears in their game wants them because they are the weird, uniquely D&D creature that is just a dumb animal with no special abilities, and having an extraordinary "ordinary" woodland beast, particular to this game, is fun and emphasizes the D&Dness of the setting. It was a mistake of the designers to designate them as anything else but beasts. As a "monstrosity" they are underwhelming; as a "beast" they are fun and make the game world more fun.
By that logic do you also include hippogriffs as beasts? They are even more stupid that owlbears (INT 3 vs INT 2), are just a mixture of two beasts: eagle and horse (similar to the mixture of owl and bear). Pretty underwhelming as a monstrosity, right?
And what about the purple wyrmling (INT 1)? Just a worm, really, huh? So, it has a poisoned stinger, but so do scorpions--so nothing new there. Sort of underwhelming also.
I, personally, include such monsters because they
ARE weird creatures, not natural by any means... so, like the designers I deem them as such, monstrosities, not beasts. They are also not common in my worlds, either.
Wait. You left out one of the three official possibilities.
It is an official possibility that the elves have already known about the owlbear existing in the material plane for thousands of years. In other words, one of the three official possibilities is, it is a natural Beast.
It is part of the second possibility, that "some fey" (those elves) insist they have always existed in the feywild...
Even if you don't want to go that route, it could be covered under the first possibility: the demented wizard created them thousands of years ago...
Either way, as I said, you do you.
