It's a region of space from where not even light can scape.

You can get a region like that by squashing a lot of mass in a small space, like happens when a star collapses under its own gravity. So here the intuition of "high density" makes sense.

But at the center of galaxies you have the so called "supermassive black holes" which are more or less comparable in size to the solar system and yes, they have a lot of mass but they are not very dense, a pop-sci trope is comparing it's density to cotton candy or even the air we're breathing right now.

So it's a matter of how you distribute mass/energy in a given diameter, not exactly of density.