If I understand correctly, your sets $A_i$ are already indexed, since otherwise you probably wouldn't be calling them "$A_i$". So, I think it would be the best to use the notation $\phi_{i,j}:A_i\to A_j$ as you have suggested. As the sets themselves are already indexed, there is no need of using any extra numbering and you can also use English names or colours or whatever you want. I shall explain in more detail below.
So, you have a collection $\mathcal{A}$ of sets. You want to give them names (you may have done this already). This is achieved by choosing some index set $\mathcal{I}$ and a function $\Phi:\mathcal{I}\to\mathcal{A}$. This has the effect of naming the sets: for $i\in\mathcal{I}$ it is now convenient to write $\Phi(i)=A_i$. (And it is this what makes it possible to consider the same set multiple times in possibly different ways. Since, if $\Phi(i)=\Phi(j)$ is the same set for two different indices, you might consider this as two "instances" of the same set. If you want consider a set multiple times in a different way, naming it by two different names is, I think, basically unavoidable.) If you don't like numbers, you can take $\mathcal{I}$ here to be a set of English names, thus giving your sets English names.
But, as you are saying, your bijections are uniquely determined by the ordered pairs of sets. So, there might not in fact be any real need of indexing your sets and you might just work with the (non-indexed) collection $\mathcal{A}$. And if you want to index it anyway, simply use $\mathcal{I}=\mathcal{A}$ and define $\Phi:\mathcal{I}\to\mathcal{A}$ to be the identity map.
I think there is no need of doing this in your case, however. Since you have stated that the bijections are uniquely determined by the pairs of sets, you might just as well index these bijections by the pairs that uniquely determine them. For example, if $\phi:A\to B$ is the unique bijection from $A$ to $B$, you might just as well name it $\phi_{(A,B)}$. As the pair $(A,B)$ uniquely determines the bijection, such notation should pose no problems and is well defined in the sense that if $A=C$ and $B=D$, then $(A,B)=(C,D)$ is the same pair and thus uniquely determines the bijection $\phi_{(A,B)} = \phi_{(C,D)}$. (Note, that in this case, you will never really need to use two names for a single set.)
I hope any of these suggestions helps.