When I was doing some research on this, I saw most guides suggest installing various xbox controller driver shims. I can easily imagine the steam controller mapping fulfilling the same role.
But yeah. I have an ultimate whatever the hell (the xbox-like with hall effect sticks) that I connect with the dongle and no issues in Ubuntu. Never tried to pair with my linux PC using bluetooth. Only issue I have is that getting the various 8bitdo software suites running in wine is a bit of a headache but there is a phone app that works more than well enough.