As you probably understand, playing a game with friends using a service online will require you to provide certain information in order to work. I'm just wary of companies/governments, other orgs collecting and using data beyond what is needed for providing that service without an option to disable it.
I ain't giving Amazon and game companies my phone number. I use protonmail, I have a Linux box and Steam Deck. I don't play games that have invasive anticheat, as in I dropped Riot Games completely.
For true privacy and anonymity (beyond what I do regularly for myself), it is best to play single player and local multiplayer games that can function without internet access. Nintendo Games all work fine with local wireless play without the need to register online. GOG and itch.io allow you to purchase games that you can keep forever. Keeping gaming installations, drives, computers separate to your productivity is another idea.
This is a legal grey area, but if you and friends purchase a game then afterward find a way to play it without the need to connect to Steam or the game publisher's servers, that is also a way to preserve your privacy/anonymity.