My question: Let $\cal{C},\cal{D}$ be categories (resp. stacks). Let $F:\cal{C}\rightarrow\cal{D}$ be an essentially surjective functor, i.e. surjective on isomorphism classes of objects. Then is $F$ an epimorphism in the category of small categories (resp. stacks)?

Answer: No. For example, any functor between categories with one object is essentially surjective, but e.g. if $M_1, M_2$ are two nonzero monoids then the inclusion $M_1 \to M_1 \oplus M_2$ of a direct summand, thought of as a functor between one-object categories, is not an epimorphism of categories.

Keep in mind, though, that "epimorphism in the category of small categories" is not obviously the "right" concept in any particular application, for multiple reasons. It discards natural transformations, so you're ignoring the fact that you are really working in a $2$-category; and there are also various notions of "epimorphism" you might want in any particular case.