Graphs are given as abstract sets of vertices and sets of edges (as adjacency lists or matrices). The problem is that corresponding vertices in the two graphs may not have the same label, so to solve graph isomorphism naively, one would need to try all n! possible bijections between the vertex sets (where n is the number of vertices).
It should be noted than one of the common used algorithms, called VF2, and descendants of it basically just check all possible bijections, although they use some heuristics to prune the search tree.
I think pictorial representations are what is confusing the grandparent. The problem isn't about comparing two graphs that look different visually, it's about comparing two graphs which have differently named labels (but might still have the same structure). The graph isomorphism algorithm doesn't know anything about where on the page you drew the nodes, just what is connected to what
