I don't understand why you are linking to something about labelled graphs when people are (presumably) trying to distinguish between graphs up to isomorphism i.e. after modd'ing out the labeling.
It doesn't relate to machine learning (which is what I assume you mean).
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.