How to make an audio 5.1 switch?
Hm, while you can technically do this with analog switch ICs, you'll have a lot of fun with ground loops, reference levels, and control.
Sounds like a job for a lot of relays, that you control with a power source connected to a rotary switch. That's costly.
Or: really just a get a RaspberryPi or similar, a decent 5.1 USB sound card to attach to that, and network cabling instead. You can get network sound infrastucture that works on basically any operating system (including exotic ones like microsoft windows) as client and on linux on the pi as server. I'd look into JACK audio.