If all black holes are by definition infinitely dense and infinitely small, how can there be different sized black holes? Isn't infinity non-negotiable?
Black holes are not by definition infinitely dense and infinitely small, so your question is a moot point.
Black holes are defined by the property of having a horizon, namely a surface that separates the outside Universe from regions from within which light cannot escape. The black hole horizon has a finite surface area, determined by the mass, rotation and electric charge of the black hole. Thus black holes can come in all sizes.
Matter in the Universe is on average electrically neutral - there being as many protons as electrons, as with atoms - so we do not expect black holes to have any significant electric charge. Thus it is mainly the mass and the rotation that determine the size of a black hole, according to the solution of Einstein's equations discovered by New Zealander Roy Kerr in 1963.
For two decades we have been measuring the mass and spin of the supermassive black holes that lurk in the centres of galaxies. Because light cannot escape from a black hole, until recently we could only determine their properties from the change of wavelength of X-rays given off by matter falling in. Such X-rays are only visible to our telescopes for the supermassive variety of black hole.
The recent discovery of gravitational waves from the collision of two black holes of 29 and 36 solar masses to produce a single black hole of 62 solar masses, and a spin 2/3 of the maximum predicted by the Kerr solution, is the first time that we have direct evidence of the millions upon millions of smaller black holes that are out there.
The mathematical solution of Roy Kerr that describes these objects has a ringlike region deep inside the black hole where our mathematical description of space and time breaks down. Formally speaking the maths is saying that the density appears to be getting infinite but what that really means is that our geometrical concept of volume breaks down - and density is mass divided by volume. Many physicists hope that a theory of quantum gravity will resolve the as yet unsolved mysteries of black hole interiors.
Professor David Wiltshire
Department of Physics and Astronomy
University of Canterbury