I think I have posted this before. Are there only 2 companies that make the GPU chips? Not like graphics cards, the chips themselves like the GK106 and AMDs Hawaii.
Do you mean designing the chips or manufacturing the chips? Primarily, you have AMD and Nvidia designing the chips, and primarily GlobalFoundries and TSMC manufacturing them.
S3 and Matrox still design GPUs, but they haven't been performance competitive (at all) in ~12 years. AMD and Nvidia are the ones you hear about because they're the ones who make GPUs that are any good for gaming. Other companies (ARM, Imagination Tech, TI, Samsung, etc.) design GPUs for integrating with mobile SoCs.
Lots of companies could manufacture GPUs, but generally don't, except as part of an SoC. Samsung, Intel, etc. If Nvidia or AMD thought it was worth their time and money to design for someone else's manufacturing process, they would.
Design of the chips.