TITLE: Correct notion of "sameness" for objects in a category
QUESTION [12 upvotes]: I want to compare two notions of "sameness" for objects in a category $\mathbf C$:

Isomorphism. This is the standard notion of "sameness" in every textbook that I know of.
Two objects $A$ and $B$ are "equivalent" iff there exists an automorphism $F$ on $\mathbf C$ (that is, a functor $F:\mathbf C \to \mathbf C$ where $F$ is an isomorphism in $\mathbf{Cat}$) that swaps $A$ and $B$ while leaving everything else unchanged. That is, $FA=B$, $FB=A$ and $F\left(O_1\xrightarrow f O_2\right)=O_1\xrightarrow f O_2$ whenever $O_1$ and $O_2$ are not $A$ or $B$.

I think the "equivalence" in (2) makes more sense for me, because it generalize the notion of isomorphic objects (it is easy to show that isomorphic implies "equivalent"), while it still preserves cat-theoretical properties. Here is an example demonstrating this:

Clearly objects $A$ and $B$ are not isomorphic, but they are "equivalent". Intuitively, $A$ and $B$ are indistinguishable within the category.
So I am confused why the notion of "equivalence" in (2) is not widely studied and not even defined in any materials I've read.
Also, while only isomorphic $\implies$ "equivalent" is true in general, I was wondering if the other direction holds for most categories that is useful and natural (unlike the artificial example I give above).
EDIT
Having read the answers and comments, I came up with some new thoughts.
Previously, I tied the notion of "sameness" to objects that have the same cat-theoretical properties, i.e. $A$ and $B$ are the "same" in category $\mathbf C$,  if for any well formed formula $\Phi(v)$, $\mathbf C \models\Phi(A)$ iff $\mathbf C \models\Phi(B)$. This is why I thought $A$ and $B$ in the previous example should be the "same".
Now I tend to believe "sameness" should work just like the usual "equality", so it should satisfy more axioms, such as how substitution would work for "equality": 
If $A$ and $B$ are the "same", then for any well formed formula $\Phi(v_1,v_2)$, $\mathbf C \models\Phi(A,A)$ iff $\mathbf C \models\Phi(A,B)$.
Let $\Phi(v_1,v_2)$ be $\exists f((dom(f)=v_1)\wedge (cod(f)=v_2))$, then $A$ and $B$ cannot be the "same" in the previous example since $Hom(A,B)$ is empty while $Hom(A,A)$ is not.
My conclusion: Isomorphism is the correct notion of "sameness" because

It behaves just like the usual "equality".
It is modeled after mathematical practice in topology and algebra, etc.

REPLY [2 votes]: This is only a partial answer, but maybe it belongs in the answers rather than the comments. In the comments, @jgon gives an argument that if $A \cong C$ for some $C\ne B$, then exchanging $A$ and $B$ via your functor $F$ implies that $B \cong C$ and therefore $A\cong B$.
This suggests to me that your notion of equivalence is more naturally defined on collections of objects rather than objects. Suppose we have $A\cong C$ and also $B\cong D$, and that $A\ncong B$. It may be the case that there is an isomorphism of categories that exchanges $A$ with $B$ and also exchanges $C$ with $D$, while leaving everything else unchanged. In this case we might want to say that the sets $\{A,C\}$ and $\{B,D\}$ are equivalent to one another. This makes sense and feels reasonably natural to me - we can't exchange $A$ with $B$ because they belong to different isomorphism classes, but we can exchange the two isomorphism classes.
This feels to me like it might sometimes be a useful notion. For example, suppose I have a category of two-dimensional shapes whose morphisms include rotations and maybe some other rotationally-symmetric operations, but not reflections. Then I might end up with two distinct classes of isomorphic shapes, where for each object in one class its mirror image is in the other. (For example, one might be all possible rotations of an L-shaped Tetris piece and the other might be all possible rotations of a J-shaped one.) Your definition might tell me that the two classes are equivalent to each other, which seems a reasonable thing to want to know.
(But then again, to make that work, we might have to consider an endo-isomorphism that swaps all objects with their mirror image, not just the ones in that particular pair of equivalence classes. This would take us even further from your idea.)
In any case, this suggests that your idea is best seen not as an alternative to isomorphism but as a higher-level notion that allows us to talk about equivalence between equivalence classes.