I would say Penrose is a mathematically literate physicist and I don't think he would consider himself to be 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 cosmology. 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.
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.
Theoretical physicists can generally only win the Nobel Prize when there is undeniable evidence from experiment of the correctness of their theoretical work. 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 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'.
Finally, Nobel Prizes are not awarded for bodies of work in the way that a Fields Medal is. The Prize is not a statement on the correctness of any of his other work in physics (some of it outside of relativity being very controversial). The Prize was awarded only for his theoretical work predicting black holes. For example, Einstein was awarded the Nobel Prize only for his work on the photoelectric effect, and the awarding of the prize had nothing to do with his work on general relativity.