@CP3 Are you able to use any other USB peripherals with your phone? Do you get any other connection options by tapping on the charging notification?
This is what I see when both plugged in to charge and plugged into the controller.
@CP3 Ah, I assumed you were referring to the charging notification when you said that you could see that the phone was charging the connected controller.
When I connect the Stadia controller to my Pixel 3, the white light on the controller comes on and the phone immediately asks if I want to let it automatically launch the Stadia app. There is no charging notification displayed, but if I search for "USB Preferences" in the phone's Settings it shows that the USB connection is controlled by the connected device (the controller) and I'm not able to change any of the "Use USB for" options. If I turn off the controller by holding the Stadia button for a few seconds I do get a charging notification, but the USB Preferences screen still shows the same configuration. So it certainly sounds like something is not working right on your end.
Apologies if I missed it, but are you able to use any other USB peripherals with your phone - USB card readers, headphones or headphone adapters, mouse, keyboard, etc? Or can you test the controller plugged in to a computer? Does your controller's light come on?