I believe that the question should be the other way around:

Why is it that you have to lose your anonimity when you are on the internet? The real world always allowed that until it became dependent on surveillance capitalism. Of course you need to prove you're yourself for some things, but that should be the exception. You could always look things up at your local library while being anonymous (for checking out you'd need a card), you could call from a payphone while being anonymous, you could use coins (cash in general) while being anonymous.

Anonimity was the rule and should still be the rule