Philosophical questions concerning the difference between equality, isomorphism, equality upto (unique) isomorphism, undistinguishability, and the like are not very popular among practicing mathematicians: there's a difference between equality upto isomorphism and equality upto unique isomorphism, and that's it (not to forget about isomorphism and natural isomorphism).
But personally, I'm not totally satisfied with this stance when looking at truly abstract structures like unlabelled graphs (finite or infinite, countable or uncountable), conceived as nothing-but-dots-and-arrows.
Are two abstract structures not to be considered equal in the strongest sense - being one and the same - as soon as there is an isomorphism between them, regardless of being unique, natural, and/or not?
The existence of isomorphisms in turn tells us something about the symmetries of the abstract structure, but of one and only one.
Maybe there are no such abstract structures per se, but only concrete structures (models) and/or concrete presentations of them (like adjacency matrices). Then the question misses a subject. But if so?