The case when $P=Hom_C$ can also be captured as $[\mathbb{N},C]$, in which case it has all the limits and colimits that $C$ has. Do you have other special cases of interest?

arxiv.org/abs/2205.00127 seems relevant

Good point. Luckily either of the options you mention results in a monad. To get the correct Kleisli category, you'll need to send the empty set to itself.

It's the Kleisli category of the monad that sends a set to the set of all non-empty subsets of it ("non-empty powerset monad").

after a bit of digging, I discovered that a sufficient condition based on similar factorization properties is proven in A. Pultr: Isomorphism types of objects in categories determined by numbers of morphisms. Acta Sci. Math. Szeged35(1973), 155–160