Equality can be philosophically troubling, but as a practical matter, what you care about is the indiscernibility of identicals, and if every property you use follows from a universal property, then equal up to isomorphism is just equal. So every Rubik's cube is equal considered as a group of face rotations, and there's only one Rubik's cube, but if you care about chemical properties of sticker glue, they're no longer equal.