The only obvious thing I'd make sure of is that you're actually plugging the headphones in *all the way* until it clicks, the port is a bit stiff.
For now, it's more of a user support forum.
If you've tried multiple headsets/headphones and the problem persists across each of them then it sounds to me like it's worth contacting Stadia support directly about it.
https://support.google.com/stadia/?hl=en-CA#topic=9634925&contact=1
Use the "Contact Us" at the top right to get started and good luck.