Can we add a structure notion on set relations without invoking choice?

Question

To $\sf ZF$, can we add a primitive one place function symbol $\mathcal S$ [read as the structure of set relation _] such that for any binary relations $Q,R$ that are sets, we have: $$1. \ \ \mathcal S (Q)= \mathcal S(R) \iff Q \text{ isomorphic to } R \\ 2. \ \ \mathcal S(Q) \text{ isomorphic to } Q$$; without invoking any kind of choice?

Answer 1

No. Not even for the empty structure, that is for sets and equipotence. See:

Pincus, David, Cardinal representatives, Isr. J. Math. 18, 321-344 (1974). ZBL0302.02021.