Assume the existence of a bijective correspondence between the poset of the subobjects of $C$ and that of $D$. Is it true that $C$ and $D$ are isomorphic?
In the case of the category of sets, this is true (just work with the singletons to define the bijection). In this way we can reconstruct the isomorphism via the bijection of subobjects. Is it possible to do something similar in a generic category?
No. For a simple example, consider a category with two objects and only the identity arrows. The two objects both have a single subobject, but they aren’t isomorphic.
Furthermore, it is not necessarily true in ZFC that a mere bijection between power sets implies a bijection between the underlying sets. For instance, if Martin’s Axiom holds and the continuum hypothesis fails, we find that $2^{\aleph_0} = 2^{\aleph_1}$. If, however, we posit an order isomorphism between $P(A)$ and $P(B)$, we can show this is induced by a $B \to A$ using the singleton set argument you mentioned.