New iPhone 16 logged me out of Instagram and FaceBook

Google Authenticator has a sequence needed to relocate to another device.

https://support.google.com/accounts/answer/1066447?hl=en&co=GENIE.Platform%3DAndroid

https://www.howtogeek.com/425994/how-to-move-google-authenticator-to-a-new-phone-or-multiple-phones/

if you still have your old device, or if your codes are stored at Google, you can use that both to log in here, and to transfer the authentication data to your new iPhone.

Otherwise, you will need to discuss account recovery with Meta Facebook support.

And once you regain access, request your data download from Meta Facebook, and get your data stored and backed up elsewhere.

For multi-factor authentication without using the Google Authenticator app, the built-in Apple Authenticator can use iCloud to store its data, which means it’ll work across all Apple devices associated with the Apple Account.

You'll need to contact Meta Support for help with their Apps and their log in process.

For the authentication App, it would be whatever app you set up originally for it. As it needs to already have your IG account in it to generate the code for it.

https://help.instagram.com/566810106808145

https://www.facebook.com/help/132694786861712/