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
Signal is free and open-source. It cannot be denied that basically everything, including minor details like usernames, is end-to-end encrypted and kept secure. The Signal protocol has been proven to be secure by many independent experts and thus it is mathematically impossible for Signal to gain access to your sensitive information (except for your phone number, obviously).
A phone number alone just won’t do much.
Signal is not open source, its a centralized US service, and you have no idea what their server is running. They even went a full year without publishing server code updates at one point, until it caused enough of a backlash that they started doing it again. But publishing that is no guarantee of anything, because you have no access to their server.
A phone number in most countries, including the US, means your real name and address.