I had the same problem with my Nintendo switch as I wanted to listen to my switch and my PC at the same time. You can setup a pulse audio loopback device. It works, but overall I hated it, because sound quality did deffinitely suffer as far as the switch goes. There where a lot of bars to figure out where to put (to not make it suck as badly) and I had to do all that twice, because I also use Windows on the same PC. By the way the results on Windows that I got always where plain better (but activating and deactivating that all the time is also annoying). Now I have the big knob the person above me already mentioned and a USB dac for the switch… Bit pricey for the task. But it did solve my problem in a way that I dont get insane with.
This was the alias I put into my bashrc to activate that loopback device
alias switch='pactl load-module module-loopback' so
pactl load-module module-loopback is what you gotta run. I´m currently not sure if there was any setup prior to that involved. I think that´s all you gotta do (thought might be wrong) it should show up in pulse. I did not really want it to be there on boot. So I did run my little
switch alias everytime I wanted to use it. From then on out I wish you good luck tweaking it.