OK. I got it. I found the error (for myself).
After I figured it out (however), I was able to reproduce the error today.
After I only paired the iPhone via Bluetooth (disable CarPlay) to synchronize my contacts with the car, I couldn't start CarPlay again.
After I deactivated Bluetooth on the iPhone and only WIFI was active, I was able to start CarPlay again. After CarPlay was connected, I activated Bluetooth again on the iPhone and it continued to work.
Maybe it will help you to completely deactivate Bluetooth in the CarPlay pairing process.
Good Luck