Multiple choice is equivalent to AC in ZF

Fix $A\subseteq V_{\theta }$ with $\theta$ limit and let $k=h(V_{\theta })$. By the hypothesis fix a single well-order $W$ of $\mathcal{P}(k)$. No sequence of arbitrary well-orders is chosen.

Define well-orders $W_{\beta }$ of $V_{\beta }$ for $\beta <\theta$ recursively. At zero use the empty order. Given $W_{\beta }$, its unique order isomorphism $e_{\beta }:V_{\beta }\to {\eta }_{\beta }$ has ${\eta }_{\beta }<k$, since $V_{\beta }\subseteq V_{\theta }$. Order $V_{\beta +1}=\mathcal{P}(V_{\beta })$ by transporting the restriction of $W$ along $u\mapsto e_{\beta }[u]\in \mathcal{P}(k)$.

At a limit $\lambda \le \theta$, order elements first by their least stage of appearance below $\lambda$, then within the same stage by its already constructed order. Every nonempty subset has a least appearance index and a least element at that index, so this is a well-order. This defines the limit rule and also the final order on $V_{\theta }$. Restrict it to $A$.

For any family of nonempty sets apply the result to its union and select the least member of each set. The empty family uses the empty function. This proves AC from the powerset hypothesis; MC supplies that hypothesis. Conversely AC supplies a point in each nonempty set, whose singleton is a multiple selection.