This question is a follow-up from my recent question, Classifying set theories whose standard models sharing the same ordinals are equal

Let's say that a (recursively axiomatizable) set theory $T$ extending ZF is "$P(Ord)$-categorical" if, whenever $M$ and $N$ are transitive models of $T$ sharing the same sets of ordinals, one has $M=N$. For example, if $T$ proves the Axiom of Choice, then $T$ is $P(Ord)$-categorical, by Theorem 13.28 of Jech's {\it Set Theory}. Is the converse true? That is, if $T$ is $P(Ord)$-categorical, then must $T$ prove the Axiom of Choice? Or, alternatively, perhaps somehow there is a consistent extension of ZF + Axiom of Determinacy, for example, that is $P(Ord)$-categorical?

If a theory's $T$ being $P(Ord)$-categorical is not equivalent to $T \vdash $ Axiom of Choice, is there an axiom or axiom schema $A$ such that $T$ is $P(Ord)$-categorical if and only if $T \vdash A$ for all recursively axiomatizable extensions $T$ of ZF?