The device name set on the Sony | BRAVIA Connect app is not displayed in the AirPlay device list
Due to AirPlay specifications, the device name displayed will be the model name of your device, regardless of the name set in the Sony | BRAVIA Connect app.
If you want to connect BRAVIA Theater Bar 9 to AirPlay, for example, you can just select BRAVIA Theater Bar 9 as the model name and go ahead with the settings.