I recently came across this paper, where the author (in section 3.1) gives an interpretation of ZFU within ZF. The author of the paper goes on to show what he calls the "synonymy" (i.e., that ZF and ZFU are isomorphic in $INT_0$) of ZF and ZFU. What is the significance of this "synonymy" of ZF and ZFU given the meta-theoretic differences between ZFC and ZFCU (some of which were outlined in my question and the answers to it)? Does a similar "synonymy" claim hold between ZFC and ZFCU?
Does it show something about the significance of adding the Axiom of Choice? Does it give us reason to take the meta-theoretic differences "less seriously" (i.e., to think that future proofs will close this meta-theoretic gap between ZFC and ZFCU)?