You are right. Both the Big Bang and a black hole are what is called a space-time singularity and the physics of these object is, to put it simply, poorly understood. We know a lot about a black hole seen from outside, or about what happened some ridiculously small after the Big Bang, but it is unknown if the laws of physics as they are currently understood cease to be valid before the Planck epoch ($10^{-43}$ seconds after the Big Bang).
The same applies for a black hole: laws of physics describe extremely well what happens outside the black hole, and, to an extent, even in proximity of the center of a black hole but the closer you get to the singularity, the higher the curvature and we simply don't know if Einstein field equations (the cornerstone of General Relativity) are valid for higher curvature, or are just a low-curvature approximation or the exact laws.