In the real universe, black holes contain probably don’t contain physical singularities that we know of. They would be light-like in a charged rotating one so that will pose a problem too.
Physics in general tends to use Perturbation Theory. You start with a simplified problem that you can solve exactly, then you move on to the next “perturbed” or solvable chunk by adding terms then…rinse and repeat as needed.
The singularity in a black hole is just the point were the mathematics of a known incomplete theory breaks down. In this case when you reach the “singularity” differential geometry fails and the entire premise of the theory are violated. The theory just cannot make any predictions at that point.
At that point there simply is no theory and the GR makes no predictions about physics there. Contrast this with the singularity at the start of time. Penrose-Hawking theories show that the start of time singularity is geometrical and physical, not topological.
Another important part here. Observers outside the event horizon will never see the singularity never form even if they could break the rules and see inside. Because from an outside observer it just won’t exist in a finite amount of time. That central point source singularity is a limitation in the current math that will be reach at a finite amount of time for someone who does fall in past the event horizon.
Singularities are problems for doing the math and math is the language of physics so they are very real for those who are working on these problems.
It is not safe to assume that the singularity is a literal physical things or places or that they are even similar in the challenges they pose.