Let's assume the merger of a binary black hole and consider especially the moment of the transition from the last stable orbit to the merger, i.e. the transition where two black holes form one black hole and thus two singularities one. Here I‘m not sure if it makes any difference if we discuss mathematical black holes (i) or physical black holes (ii) (physical in the sense that the mass is not contained in a point and hence the break down of GR is avoided). While in (i) the singularities which contain the masses are a point in time instead in (ii) the masses are part of the manifolds (this is just my guess I can be wrong). Are in the latter case the two masses at two different „locations“ before and instantaneously(?) at one location after the merger? In general how would you describe and eventually distinguish these two cases?
in (ii) [physical black holes] the masses are part of the manifolds (this is just my guess I can be wrong)
Yep, you were wrong :-). In any black hole, we can't really localize the mass. The difference between a Schwarzschild black hole and an astrophysical one is that a Schwarzschild black hole has always existed. It didn't form by gravitational collapse.
While in (i) [a Schwarzschild black hole] the singularities which contain the masses are a point in time
The singularity is spacelike in both cases (which makes it similar to a spacelike surface, which is similar to a point in time).
In general how would you describe and eventually distinguish these two cases?
Basically there is no interesting distinction between the two cases with respect to a black hole merger. An astrophysical black hole differs from a Schwarzschild black hole in the past, when they formed. The merger happens after they have already formed.