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?
In practice, God does not hand you abstract structures: you construct them from other structures, and the point of caring about isomorphisms is that you want to keep track of this construction process. For example, you are almost never handed a vector space $V$ and its dual $V^{\ast}$. Usually you are performing some construction (e.g. on the tangent space $T_p(M)$ of a manifold $M$ at a point $p$) which naturally involves elements of $T_p(M)$ as well as elements of the cotangent space $T_p(M)^{\ast}$, and if you are foolish enough to think that they are the same space then you will literally not be capable of doing calculations in this setting (e.g. changing coordinates).
One way to say this is that abstract structures often arise functorially, and even if $F(c), G(c)$ are isomorphic where $F, G : C \to D$ are functors and $c \in C$ is an object, the functors $F, G$ need not be naturally isomorphic, and usually we actually care about the functors, not the objects $F(c), G(c)$ in isolation. In the above example taking duals is a contravariant functor $\text{Vect} \to \text{Vect}$, and since it is contravariant it does not behave at all like the identity functor.