When the car or phone get a software update, they both usually need to be unpaired and removed, then re-added as Media Bluetooth options because it seems to lose the address.
I'm using a Pixel 9 Pro XL and there was an Android Update "Baklava" 16 and would not handshake Media Audio via Bluetooth and did not recognize via USB.
Removed phone from car. Removed car from phone.
Re-paired both.
Left the vehicle and let it sleep/lock.
Came back and it works.
(If not, soft-reboot finishes the process)
I've had to do this every time an update via Phone or Car occurs. This method always works for me.