Does Babai's quasipolynomial time $\mathsf{GI}$ algorithm actually generate the isomorphism?
These problems are polynomially equivalent.
Indeed, suppose that you have an algorithm that can decide whether two graphs are isomorphic or not, and it claims that they are.
Attach a clique of size $n+1$ to an arbitrary vertex of each graph.
Test whether the resulting graphs are isomorphic or not.
If they are, then we can conclude that there's an isomorphism that maps the respective vertices to each other, thus we can delete them.
By repeating this test $n$ times, we can find (a possible) image for any vertex.
After this, we attach another clique, this time of size $n+2$ to a (different) arbitrary vertex of each original graph, and proceed as before, etc.
Eventually, we'll end up with two graphs that are isomorphic, with cliques of size $n+1,\ldots n+n$ hanging from their vertices, which makes the isomorphism unique.