I suppose OP defines it as the mass of the BH, divided by the apparent volume taken up by the BH (more precisely: the apparent horizon), as seen from the outside. Put differently, for a Schwarzschild BH: Density ~ M/R³ (modulo constant prefactors) ~ 1/M², since the Schwarzschild radius is linear in M.

Density is mass divided by volume. Volume of BH is defined by event horizon.

Singularity is a result of applying a theory of relativity in a case where we know relativity doesn't work, so it's unlikely it's real.

> Volume of BH is defined by event horizon.

The event horizon is a three-dimensional null hypersurface, though, encompassing a four-dimensional spacetime "volume". You are probably referring to the two-dimensional apparent horizon, which depends on the spatial slicing.