They are, to the atom, exactly the same. — afterthegame

Well, first look, they cannot be. And, just where is the "identity"? It seems to me that might make a difference.

Now, can we write a definition of "identity" that allows us to treat either one of them as an individual object? — afterthegame

I am not sure what you are asking. Are the two balls identical? Yes, because that is a given. Are there two balls rather than one? Yes, that is also a given.

Is this a lead-up to questioning the principle of the identity of indiscernibles?

Let's frame this problem in the right context, Leibniz's principles concerning identity.

1. Indiscernibility of Identicals: The less controversial principle that states that if x is identical to y, what's true of x must be true of y too. My favorite example to illustrate this principle is Lewis Carroll aka Carl Lutdwidge Dodgson. Lewis Carroll is identical to Carl Lutwidge Dodgson; ergo, what's true of Lewis Carroll is also true of Carl Lutwidge Dodgson.

2. Identity of Indiscernibles: This is the principle that's harder to swallow. It basically asserts that if whatever is true of x is also true of y, x and y are identical i.e. they are the same thing.

Notice that in 1 above, Lewis Carroll and Carl Lutwidge Dodgson are the same individual and if I were to form a set with this information, I would get A = {Lewis Carroll} or A = {Carl Lutwidge Dodgson} and the number of elements in set A, n(A) = 1. In other words, identity in numerical terms is 1.

Coming to your two black balls, if I were to form a set with them, it would look like this B = {black ball 1, black ball 2} and if I were to count the number of elements in set B, n(B) = 2. However, according to principle 1 above, the only number that validly describes identity is the number 1. Ergo, the two black balls aren't identical.

This doesn't work either because there is nothing else in this universe apart from the two balls. — afterthegame

If there's nothing else in the universe, then we cannot treat them in any way, including in terms of identity. We aren't there. If we are there, we are there somewhere, then one of them is to the left and the other to the right or one is behind the other. And there's the identity. We can't get more specific since they are the same in all other ways than location. The difference between them is location. And then, they are not the same objects, since there are two of them. Without an identifier I am not sure what identity means. If they are alone in their universe, well, they are not the same ball by definition, since there are two of them.

In many situations with two of the same thing, they have no other identity than location. Like a couple of electrons, say. Perhaps there are potential energetic differences between those. But we don't need another universe to have two of the same thing. Could be two hydrogen atoms.

Yes, we can. They key is their spatial occupation. Both of the balls are defined as individuals relative to the other by position. "This ball is not that ball over there". The only time we cannot create an identity between two balls is if we say the two balls occupy the exact same physical space as well. At that point, we could not identify a "second ball", and we would identify only one ball.

Now, can we make an identity of the balls like, "That is ball 1, and that is ball 2"? That requires a human in this case, as this would be in relation to our viewpoint and preference as to which ball was which. We could say, "Ball 1 is the ball to my left, and ball 2 is the ball to my right". If we watched and the balls shifted position, we could continue to identify them. Of course, as location and our ability to identify them by their location in relation to us is the only way we could identify them specifically, we would have to constantly track them to be consistent in our identity. If we looked away and the balls moved, we would have no way of knowing which one was our original ball 1 and 2 at that point.

Now, can we write a definition of "identity" that allows us to treat either one of them as an individual object? — afterthegame

No. 10m separation identifies an unoccupied space, but does not identify the objects the unoccupied space presupposes. No definition of identity is required for the treatment of objects.

The fact that black ball is given as means for identification in that universe is irrelevant, because there is nothing else in that universe to use the identification, hence the definition of identity is not necessary in the treatment of objects.

If the claim is that an observer outside the black ball universe uses the object’s given identity as black ball, then the entire scenario is irrelevant in itself, because the observer has already identified the spatially separated objects as two black balls, which makes explicit he has already defined identity sufficiently to himself, in order to attribute “black ball” to each of those objects.

It cannot be said there are two black balls, then fail to account for them. If there is no account, it cannot be said there are two black balls. Then the whole thing becomes an exercise in irrationality.