Note: this is more of a philosophical question.
Said differently, what if we were to ask Categories to be extensional in the sense that if two objects are isomorphic then they are equal?
Is there any example where we do not want this?
And notice that this of course shouldn't make every isomorphism an identity. We still have different automorphisms.
At least intuitively I find this to follow the structuralist view of what an object is.