afaik Stadia is chrome specific. try visiting http://stadia.google.com/ on another browser. I can't even login and view my previous screenshots, let alone play a game.
Most every cloud gaming is Chrome specific. Heck the Amazon cloud gaming doesn't even work on Chrome on Linux....
You're right, I only checked with Edge forgetting that is just Chrome too.
For what little it's worth, Stadia is opening to play through Safari (on iOS) soon.