I'm going to take a different approach to this :
Macroscopic vs. Quantum worlds.
In standard model of particles it is understood that besides characteristics like momentum, spin, etc two electrons are indistinguable.
Let's remember that an elementary particle is, by definition, identified by these characteristics. The (rest) mass is a fixed value. The spin is fixed, the various charge values are fixed. If they deviate from specific values then you do not have one of these particles at all.
But we say that two e.g. electrons are indistinguishable because in any system we cannot label them and track them in any way. We can make a measurement that says there is an electron at position one and two, but the instant we make those measurements we (in general) cease to know anything about the actual positions of the electrons.
We can't say that when we next measure those electron positions which electron is which - we can't track them. The electron at position one could be at position three or four next, and the electron that was at position two could now be at position three or four. We just know the next measurements produce two distinct measurements for positions of electrons.
Are in the same sense two black holes indistinguable given they have same mass, momentum, etc?
So not in the same sense as elementary particles.
If I have two black holes they have a macroscopic position. Unlike my electrons I can track them easily and there is no mystery between measurements as to which is which. Regardless of the no hair theorem, regardless of their size, rotation, charge, etc. they have distinct locations which can be tracked.
So the black hole that started at position one, I can say with essentially perfect confidence is the same one I measured at position three later. Likewise there's no confusion about whether the other black hole could be at position three and the one I thought was there is actually at position four.
So for macroscopic objects location is a property that labels them uniquely.
But for elementary particles this is not generally the case.
The No Hair Theorem
I think I should point out that the No Hair Theorem does not say we cannot distinguish black holes from each other, even if they are identical in external characteristics. It says that the only information we can determine about the black hole's interior (on the other side of that event horizon) are these "statistical" values.