CarPlay is tricky to explain in wireless mode. And Apple and their suppliers (really there are only a handful of companies that make most audio head units like Harmon or Audiovox) also seem to be very tight-lipped about it.
From what I've been able to understand, it uses both BT and WiFi. It uses BT to do a secure handshake with the car over which your phone then connects to a hidden WiFi (i.e. nonbroadcast) network between your phone and the head unit in the car. This is a network specific to mirroring a display from the phone that we can't see but is otherwise a virtual screen (i.e. no processing in the head unit, just video decoding, display, and touch control). In the back, the iPhone 'hides' this network, as it doesn some other connections, so it's not viewable on the phone screen.
Also, my understanding is that in this specific case the phone can connect to both the car and another wi-fi at the same time.
The iPhone in particular has an odd way of handling disconnects/shutting off Wi-Fi and BT. If you use Control Center, those don't really turn them off, they just disconnect everything. To turn it off really, you have to go to Settings
How To Completely Turn Off WiFi On Your iPhone | Ubergizmo. Airplane mode from Control Center only turns off the cell phone.
Anyhow, I'm not having any issues with 15.6.1. Make sure you don't have Wi-Fi disabled in Settings (vs Control Center).