Set-Theoretic Omniscience and Set Definition
The distinction you're seeing is a very real one, and is treated seriously in both philosophical and mathematical logic; the prevalance of the extensional approach should be taken as a sign of its strength rather than the weakness/uninterestingness of intensional approaches.
You're getting at precisely the extensionality versus intensionality issue. The extensional perspective (as per the axiom) says that a set is determined entirely by its elements; the intensional view, by contrast, says that there is additional relevant data, and usually this data is taken to be some meaning or process assigned to the set.
It's important to note that this issue persists outside set theory. For a natural language example, "morningstar" and "eveningstar" are extensionally equivalent but are a priori different notions; so we can imagine an "intensional" universe of names sitting above an "extensional" universe of referents. This is a huge motivation in philosophical logic, and in particular modal logic; see e.g. here. It's also a natural aspect of intuitionism, or any philosophy of mathematics which views the mathematical universe as a "product of the mind" (or similar).
To be a bit more specific:
Your particular focus on how hard it is to "know" that two things are equal takes us to a very precise notion on logic, namely provable equality (or more generally "provable ---ness"). This is generally a hugely important idea, even when we're only interested in a semantic and extensional picture of things. I think the place this is most obvious is in Henkin's proof of the completeness theorem. Given a consistent theory $\Gamma$, Henkin (after perhaps modifying $\Gamma$ in a technical way) looks at the structure $\mathcal{T}(\Gamma)$ assigned to $\Gamma$ as follows:
Elements of $\mathcal{T}(\Gamma)$ are equivalence classes of terms in the language of $\Gamma$, with the equivalence relation being "$\Gamma$-provable equality."
The language of $\Gamma$ is then interpreted on this set of equivalence classes by saying that relations hold precisely when they provably hold: e.g. if $U$ is a unary relation symbol in the language of $\Gamma$ and $t$ is a term in the language of $\Gamma$, then $U$ holds of the equivalence class of $t$ in $\mathcal{T}(\Gamma)$ iff $\Gamma$ proves $U(t)$.
So the objects of $\mathcal{T}(\Gamma)$ are "really" just names for objects. Note that this is all taking place inside classical logic: the intensional perspective is useful even within extensional mathematics.
Conversely, one could always argue that "intensionality is extensionality in disguise" - whenever we move from an "extensional world of objects" to an "intensional world of objects-via-definitions," what we've really done is move to an "extensional world of definitions" - our new objects are definitions, and they're treated extensionally! This is of similar spirit to how many-valued logic can be situated inside first-order logic, or how classical logic can be embedded into intuitionistic logic, or ... Basically each perspective is so broad that it can simulate the other. This is a good thing for each (in my opinion).
- Another example of this phenomenon is the idea of "provable totality" in proof theory. Roughly speaking, a function is provably total relative to a given theory if it has some definition which the theory proves defines a total function: that is, the theory should prove $\forall x\exists !y\varphi(x,y)$. The actual notions of provable totality considered arise by restricting attention to definitions of certain forms, e.g. via Turing machines, and understanding what functions a theory proves are total (in a given sense) gives us lots of information about the theory as a whole. This is closely connected with a similar but more technical notion in proof theory: proof-theoretic ordinals.