My main device is the same for Zwift and Zwift Companion i.e. iPhone7. I am told by Zwift that they can never connect to each other under those circumstances. They connect OK when I use my PC as the main device for Zwift and the iPhone7 for Companion, but my PC is nowhere near my Wahoo Kickr. They should be able to connect, and it would be great if the signal/cable from my iPhone7 to my 42" dedicated screen was able to split so I could see both of them on my screen at the same time.
It does not work that way. First of all, Zwift will not run in the background. When put into the background, your rider stops. Second of all, Apple does not include split-screen mode/support on iPhones which would allow you to use two apps (i.e., Zwift and Zwift Companion app) at the same time.