I would say Penrose is a mathematically literate physicist and I don't think he would consider himself (at least not primarily) to be a pure mathematician. For example, his argument for the Penrose inequality is a plausible but non-rigorous physical argument.
The main contribution of Penrose and Hawking and the one cited was that they showed (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. This was built on by Schoen and Yau in 1983 who proved that black holes form when matter condenses into a sufficiently small region following on from their work on the proof of the positive mass theorem in 1979 - 1981. Essentially, Schoen and Yau show that if an initial data set is asympotically flat with a large mass density on a large region (large region being 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 experimental data for 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 nice cohesive reason to give the award to Penrose as one of the recipients.
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 as stated in the official citation.