Monster girls, as best as I can figure out the definition, are typical JRPG monsters made anthropomorphic with traits typical of (sexy) human females. Mythological creatures notwithstanding.
Furries are pretty similar in this regard, being animals made with anthropomorphic traits. So I guess monster girls could count as furries in this context. Granted monster girls can seem more human compared to furries, but still, since they’re based on creatures quite different from humans, I still regard them as being completely non-human, no matter how human they can look. After all, any (physical) monster or animalistic feature possessed, no matter how minor, considerably makes the character non-human by default. Such semantics are getting into the Ship of Theseus kind of argument though, which is more complicating then I’m willing to touch on…
Then again, “furry” can be very vague, even working in reverse (humans made to look animalistic). To that end, since “furry” can be such a broad term, monster girls might as well be “furries,” if only because it can be harder to argue against such an ambiguous definition then for it. Still, maybe the simplest answer is to just regard “monster girls” as an “unintentional” sub-category of furries. Not unlike the ponies of My Little Pony.