Nowadays, a majority of apps require you to sign up with your email or even worse your phone number. If you have a phone number attached to your name, meaning you went to a cell service/phone provider, and you gave them your ID, then no matter what app you use, no matter how private it says it is, it is not private. There is NO exception to this. Your identity is instantly tied to that account.
Signal is not private. I recommend Simplex or another peer to peer onion messaging app. They don’t require email or phone number. So as long as you protect your IP you are anonymous
2FA is an important security layer, if the service, after sending you the activating SMS with the code, delete your number (normal in serious services), it’s also not an privacy problem. In big us corporations on the other hand, it is, eg.Google store tour number and also probably share it, there 2FA is not an option. Instead a number, some services also admit alternatively a second e-mail account to receive the activation code, there, if you have doubt, you can use an disposable mail, so there isn’t any privacy problem.
2FA helps with security concerns, not privacy concerns. They still would have your number. Also about Google, they have one of the widest spread and utilized 2FA authentication applications out there.