I would say Penrose is a mathematically literate physicist and I don't think he would consider himself to be a mathematician.  Obtaining a PhD in pure mathematics with pure mathematicians does not make one a mathematician.  If you watch his interviews, he states that he had already started to turn to physics during his PhD under the influence of the physicists at Cambridge, and had started to focus on learning quantum mechanics and general relativity.  This is also clear from his papers and books. For example, his argument for the Penrose inequality is a plausible but non-rigorous physical argument.  He has never done anything towards a rigorous proof of the Penrose conjecture, and would not be expected to do this as a physicist.

Penrose and Hawking showed in different ways (roughly speaking) that if one makes some physically reasonable assumptions, the existence of a closed trapped surface implies that the evolving spacetime contains a black hole.  The proof that black holes form when matter condenses into a sufficiently small region was proved by the geometers Schoen and Yau in 1983 following their work on the proof of the positive mass theorem.  Essentially, they show that if an initial data set is asympotically flat with a large mass density on a large region (suitably defined), there is a closed trapped surface in the initial data.  General relativity represents an intersection where both mathematicians and physicists contribute decisively, and Penrose is on the physics side.

Also, let's be clear: theoretical physicists can generally only win the Nobel Prize when there is undeniable evidence from experiment of the correctness of their theoretical work. It does not matter how brilliant they are, how smart they are, how good they are at mathematics, or how original their mathematical work is. The concrete experimental discovery of a supermassive black hole at the center of the Milky Way by the other winners gave a great cohesive reason to give the award to Penrose as one of the recipients. It is absolutely not the case that the physics community does not care about mathematics, is not aware of the significance of the Penrose-Hawking singularity theorems, or that they neglected Penrose and tried to avoid giving him the prize for being a 'mere geometer'.