All our laws of physics break at the singularity. Mathematically, the black hole behaves as if all its mass were concentrated at a point in the center of the sphere made by the event horizon, and the extreme curvature of spacetime is also centered about that point. It's a useful thing to consider, and a basic application of intuition says that "the singularity has no volume, immense mass, and infinite density"

That notion violates much more than basic notions of density, it violates all of physics. Including the uncertainty principle.

It's also worth noting that it's not so much that "black holes violate physics" as it is that "physics doesn't make predictions about the singularity."

It's like if you were presented the function ln x and you asked what happens when x=0. It's not that 0 "breaks" the function - the function simply makes no valid predictions for x<=0.

Note that it's also possible to extend the definition of the logarithm with limits, and to extend the notion of the reals with signed infinity to get that ln 0 = -infinity. In the case of black holes, we just haven't figured out what those extensions on the math are that make accurate predictions, but for now it's still useful to consider things as "infinitely dense"