Game developers pay to use PhysX.
I have worked at both Capcom (as a developer on a game that used PhysX) and NVIDIA.
I have never heard of the kind of conspiracy like intentionally breaking games for AMD GPUs.
Obviously there is marketing and sponsorship stuff ("the way it's meant to be played"), but no game developer would intentionally break their game on AMD GPUs to please NVIDIA. AMD market share is something like 40% in discrete GPUs. NVIDIA would have to pay game companies 40% of their revenue on a game. NVIDIA is not quite THAT rich.
Some games are more optimized for one over the other, and there are many reasons for that (different driver implementations, missing functions, developers predominantly use 1 company's GPU, etc), but I have never heard of intentional breaking.