I assume they chose Windows to support the widest array of games, and the most comprehensive and timely support for things like RTX.
Stadia does use Linux, however Stadia has made an enormous number of missteps and seems very close to DOA so I'm not sure they're a good example of much.
It's an abstract backend host for a remote desktop sort of thing -- there is nothing "Windows-esque" about the experience for the end-user -- so I imagine they could easily support alternatives as the need arises. But given that Windows currently supports 100% of the top PC games, where alternatives support a small fraction of that, it's something they certainly needed to support for now.
It's definitely Windows. You do see the Windows 10 chrome on windows (sometimes the steam is there minized) and games in steam install to Program files...
Oh for sure they're using Windows to host most/all games, with a very good justification for doing so, but the way the client app interacts with the server they could just as easily have some games hosted in Linux. The point being that they are in no way tied to Windows, it's just the best option for what they're doing.
I see it as Linux gaming hostile for two reasons. Unlike Stadia, they help Windows gaming with this by using Windows. And unlike Stadia, even their client doesn't work on Linux.
Not that I like Stadia to begin with. They help Linux gaming only tangentially. But what they do actually helps in some way, rather than harms like Nvidia.