Also, I'm not sure you need stuff to always approach without ever getting there: it's not clear what happens to quantum fields at a singularity, but if you no longer have time displacement symmetry you don't get energy conservation so stuff might just cease to exist.

If you try to naively apply the same procedure you would use to define volumes outside the event horizon, it's proportional to the lifetime of the black hole (infinite in the classical case) for the reason you note.

Of course the better way to interpret this is probably just that the interior volume of a black hole is ill-defined.

Every geodesic that crosses the horizon terminates at the singularity in a finite amount of time. If you fall past the horizon the singularity is in your definite future and in a finite amount of proper observer time your geodesic will end, at which point ... who knows what happens.