The point is that if you are interested in the structure of finite sets, then there are structure-preserving isomorphisms between the vertices of a triangle and the set {A, B, C} so that (for example) permutations of the set and reflections/rotations of the triangle are coincident.

But if you're interested in the structure of plane geometry then there is no such structure-preserving isomorphism because any map into a finite set will "delete" information about side length and angles.

The structuralist idea is that interesting mathematics can be "most easily" found by considering structure-preserving isomorphisms and not the structures themselves. In particular dealing with the structures directly can obscure the mathematics you are trying to discover, e.g. dealing with a full Euclidean group when the dihedral group is all the problem requires.