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](http://ncatlab.org/michaelshulman/show/What+is+a+2-topos) 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).

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](http://ncatlab.org/nlab/show/generalized+universal+bundle)).

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.