It is not.

As a matter of fact, the archetypal black hole solution, the Schwarzschild solution, is a vacuum solution of Einstein’s field equations of general relativity. The density of matter is zero everywhere.

This solution can be seen as the mathematical limit of matter collapsing under its self-gravity, a situation first investigated by Oppenheimer and Snyder in 1939. Their solution does contain matter. But at any finite time, no matter how distant in the future, the density of matter in this solution is always finite everywhere. It’s only in the mathematical limit of future infinity, which lacks physical meaning, that this solution asymptotically approaches the Schwarzschild solution.

In short, no physically realizable black hole solution is infinitely dense anywhere.