Anytime where knowing a permutation of thing1 is the same as thing2 allows you to treat both thing1 and thing2 in a similar fashion. A related problem in canonization, finding a normal form you can put both thing1 and thing2 into knowing that if the normal form is the same they are isomorphic.

Compiler optimization, generating hardware layouts of circuits, and helping neural nets scale. Brute force isomorphism (where you check permutations but don't care if the object under study is a graph) is a bit lower than the n! naive method Babai uses as a subroutine in the paper, https://oeis.org/A186202