What does Aluffi mean here? On page 487 of Algebra: Chapter 0, Paolo Aluffi has a footnote referenced in a paragraph about isomorphic objects. The sentence which references the footnote reads,

The structure of an object in a category is adequately carried by its
isomorphism class [2], and a natural notion of ‘equivalence’
of categories should aim at matching isomorphism classes, rather than individual
objects.

The footnote is

[2] One should not take this viewpoint too far. It is tempting to think of isomorphic objects
in a category as ‘the same’, but this is also problematic and does not lead (as far as we know) to
a workable alternative. For example, some objects in a category may have ‘more automorphisms’
than others, and this information would be discarded if we simply chopped up the category into
isomorphism classes, draconially promoting all isomorphisms to identity morphisms.

What exactly does he mean here? I'm particularly interested in what he means when he states that some objects will have 'more automorphisms' than others because I am interpreting this in the context of two isomorphic objects, but this doesn't make too much sense to me because it seems straightforward that if $A\cong B$, then $\mathbf{Hom}(A,A)\cong\mathbf{Hom}(B,B)$.
The only interpretation that seemed plausible was that if you looked at the two isomorphic objects in a category with more structure, then their hom-sets would not be isomorphic in that category, but I'm not sure that's right.
So, does someone have a good idea of what he is referencing in this footnote? Thanks.
 A: I think the important part here is “draconially promoting all isomorphisms to identity morphisms” (emphasis by me).
We want not only to know that two objects are isomorphic, but we also want to know in what ways they are isomorphic.
If we were to pretend that ‘we identify every object $A$ with every other object $B$ that is isomorphic to $A$, such that every isomorphisms from $A$ to $B$ becomes the identity $\mathrm{id}_A$’ (which is not even possible), then we would in particular regard every automorphisms of $A$ as simply $\mathrm{id}_A$. So we would lose all informations about the automorphism group of $A$.
However, it should be pointed out that a similar looking construction does work:
Let $\mathcal{A}$ be a set of representatives for the isomorphism classes of objects.
For every object $B$ there exists a unique object $A$ in $\mathcal{A}$  isomorphic to $B$.
Be can identify $A$ with $B$ by choosing one specific isomorphism $φ_B \colon B \to A$. (For this, we belive in the axiom of choice.)
Under the induced map
$$
  \mathrm{Hom}(B, A) \longrightarrow \mathrm{Hom}(A, A) \,,
  \quad
  ψ \longmapsto ψ ∘ φ^{-1} \,,
$$
this isomorphisms $φ$ ‘becomes’ the identity $\mathrm{id}_A$.
But every other isomorphisms from $B$ to $A$ does not become the identity; in fact, any two different isomorphisms from $B$ to $A$ give different automorphisms of $A$.
So we have lost no informations about the ways in which $B$ and $A$ are isomorphic.
