When matter is dropped into a black hole, does the content of the matter impact the structure of the singularity? That is to say, other than the change in mass, do any of the singularity’s properties change?
A black hole singularity, in the description offered by general relativity, doesn't have any structure. It's a hole that's missing from spacetime, not some stuff that exists, spread over some region of spacetime. We can't say whether it has a particular size, or even how many dimensions it has, because the mathematical machinery of measurement breaks down. So since it doesn't really have any structure, it doesn't make sense to talk about whether that structure changes due to accretion of infalling matter. Even the mass of a black hole is not really a property of the singularity; it can be expressed mathematically as a property of the surrounding space.
Your titular question and the first question in the body don't make sense according to the model; a singularity doesn't have a structure and so can't have a homogeneousness. To answer your final question:
That is to say, other than the change in mass, do any of the singularity’s properties change?
Yes. You already know that dumping a few planets into the black hole will increase its mass. There are two other properties that you can change by throwing stuff into a black hole:
- Making lots of spinning stuff spiral into the black hole will increase its angular momentum. Spinning black holes can have very strange properties.
- Firing a stream of protons or electrons (or other charged particles) into the black hole will charge it. Charged black holes can also have strange properties, though we're not quite sure whether it's possible for a black hole to be charged enough for some of them to occur.
If a black hole is spinning and is charged, you need the Kerr–Newman Metric to model it. This is a generalisation of lots of other metrics; if you set the charge and spin to 0 you get the same results as the Schwarzschild metric.
A singularity is not a point property but a macroscopic property of a connected region.
For example, when integrating analytic functions along a circular path, the "basic" result is zero but the actual result depends on which singularities are enclosed in which direction. The integration becomes undefined when actually crossing singularities but their dependable behavior comes about when you don't.
In a similar vein, there are no "point charges" or "continuous currents" but the respective surface and line integrals capturing electric and magnetic fields deliver results consistent with them.
In a similar vein, a black hole's environment is consistent with it having a certain mass, and that changes summarily as mass continues to go there. In terms of being mostly indirectly observable through integral properties of the singularity's environment, it shares some annoying philosophical traits of quantum particles.