Remember that $\mathbf{2}$ is (classically) the subobject classifier in Set, so that there is a natural bijection $\mathrm{Sub}(X) \cong \hom(X,\mathbf{2})$ given by pulling back along $\mathrm{true} \colon \mathbf{1} \hookrightarrow \mathbf{2}$. So you can think of a subset of X as either an isomorphism class of monos into X or a function $X \to \mathbf{2}$.
Mike Shulman's pages about 2-toposes say that there is no subobject classifier in Cat. I don't know of a reference, but you could drop a query box there asking. Such a thing would classify functors that are faithful and injective on objects (i.e. the monos in Cat).
Edit: I misinterpreted Mike's remark at loc. cit., but see Michael Warren's answer.
The closest thing I know of (see loc. cit.) is to classify discrete opfibrations with an equivalence between functors $A \to \mathrm{Set}$ and discrete opfibrations over A, given by pullback along the forgetful functor $\mathrm{Set}_\bullet \to \mathrm{Set}$ out of the category of pointed sets (see here).
These don't look very similar, but in fact the Cat example is related to the comprehensive factorization system (initial, discrete opfibration) in Cat (see Street--Walters, The comprehensive factorization of a functor, Bull. AMS, (5)79, 1973), while the corresponding system in Set is (epi, mono). The latter also arises as the image factorization system, while in Cat, as far as I can tell, the two notions don't coincide.