The term "labeled graph" just means a graph with each node labeled differently (but arbitrarily). It just allows for reasoning & enumerating the vertex set. It's a typical assumption to make in the context of graph isomorphism.
It doesn't relate to machine learning (which is what I assume you mean).
Isomorphism of a pair of graphs usually refers to isomorphism of their unlabelled equivalents.
Yes the concrete expression of the isomorphism would be as a mapping between the labels.
Given that the paper linked to is by Brendan McKay et al, it seems reasonable to mention that nAUTy works by finding (efficiently) all permutations of the labellings that result in an automorphism of the graph.
It doesn't relate to machine learning (which is what I assume you mean).