This problem is not exactly with Stadia, but mostly with Ubuntu 18.04 and Google-Chrome.
I have a Switch Pro controller connected with my computer with Bluetooth, and my Firefox recognize it fine (tested at https://www.html5gamepad.com) but google-chrome don't.
Since Stadia (unfortunately) only works on Chrome I was unable to play using my gamepad (it doesn't recognize any gamepad connected).
Anyone has any idea on what should be the problem?
@ThOR27 Yep, sorry, I misread your initial post thinking you were talking about the Stadia controller rather than the Switch controller (which you did clearly state - I just haven't had enough coffee today).
What version of Chrome are you using, and how did you install it? You might want to reinstall from here just be sure: