I had this problem and Apple customer support was no help. After about 2 days of Googling information…
you have to be sure you have “Accessories” turned on under Settings/Face ID & Passcode. The phone has to do an update for the device.
If you plug in the adapter when this isn’t turned on, it won’t read it. It’s as if it’s not plugged in. You can tell if this is your problem by going to Settings/General/About. The Adaptor information should show up in the WiFi section. Mine appears below “Carrier Lock”. If it doesn’t show up, the phone is not recognizing the AV adaptor.
Do a hard reset on the phone. Volume up, volume down, press and hold the on button on the right until the phone shuts down, the apple icon comes on and the phone restarts.
Now make sure “Accessories” is turned on and plug the AV adaptor in. The upgrade should process for the device and the phone should recognize the adaptor.
Good luck! Figuring this out and getting it to work on my iPhone 13 when Apple support kept telling me it was my TV was quite annoying!