I'm beginning to wonder whether "support for 2FA" is a way for companies to get your telephone number into their database. Does using an authenticator application also provide the same information to the company?
No: if you use (for example) Google Authenticator, your phone need never connect to network. The QR code you scan contains all that is needed to generate the code given the current time. Older versions of the Authenticator app didn't even have network access permissions; newer ones do because Google added the ability to automagically set up a Google account for 2FA.
