In set theory is there a standard witness that two sets are "the same?"
Something like
$$ A \sim B = \{ \emptyset \mid A = B \} \cup \{ \{ \emptyset \} \mid A \ne B \}$$
Feels very ad-hoc
Something like the set of identity isomorphisms feels perverse
$$ A \sim B = \{ f \in A \cong B \mid \forall x \in A. f x = x \} $$
The notion of witness is applied to existential sentences: a witness for $\exists x \phi(x)$ is a specific $c$ such that $\phi(c)$ holds (in the context of some model, or as a step in a proof).
Now, in ZF, $A=B$ is equivalent (by the Axiom of Extensionality) to a universal sentence, namely $\forall x (x\in A \iff x\in B).$ Since the quantifier here is $\forall$ rather than $\exists,$ the idea of a witness doesn't seem useful.
Contrast this with the negation $A\ne B.$ This is equivalent to the existential sentence $\exists x ((x\in A \land x\notin B) \lor (x\in B \land x \notin A)).$ So it makes sense to ask for a witness to $A\ne B;$ this would be some set $c$ such that $(c\in A \land c\notin B) \lor (c\in B \land c \notin A).$
If you rig up some artificial way to view $A=B$ as an existential sentence (which is essentially what OP would be doing with the examples called "ad-hoc" or "perverse" in the question, if you wrote them out), it's not going to reveal anything interesting.