Display and Bluetooth

Have you put the stereo into discovery/pairing mode? Devices will not appear in Bluetooth until you turn Bluetooth on for the iPhone and then put the device you wish to pair with into discovery/pairing mode. Then both of them are looking and you should be able to pair them.

As far as looking in Media & Purchases, I see Sign Out, Password Settings, View Account, and Cancel. Are you logged into the iPhone XS Max on that device, or did you log out? Did you log into this device in iTunes & App Stores?

Can you please explain what you mean that an iPhone 13 Pro is displayed with XS Max. Where are you seeing this information?

What are you trying to sync via Bluetooth? Only certain tasks can be performed via Bluetooth on the iPhone.

You're welcome.