I have had the same issue on both versions of iOS 13. It only happens with Apple Music on CarPlay. When making selections on the "For You" tab in CarPlay, I just get a spinning wheel, and no selection is made. This is particularly frustrating because one of the most useful features of Apple Music is to find music for you based on your likes.
If I pick up my phone, I can make a "For You" selection in Apple Music without a problem and it will play on CarPlay, but that defeats the purpose of CarPlay. I need to be able to use the CarPlay screen in the vehicle. This is the only thing that is not working in CarPlay, but it was fine on iOS 12. Apple Music does not have this issue on my iPhone, my iMAC, or my MacBook Pro, so it's not an issue with Apple Music. It only happens in CarPlay.
I have turned off and restarted Bluetooth, Siri, and iCloud Music Library. I have also uninstalled and reinstalled the Bluetooth and CarPlay profiles in the vehicle. I also reset the Network Settings. Nothing solves this issue for me.