The EFF wrote, "The Senate Commerce Committee is set to consider S. 737, the SCREEN Act, ... will force millions of adult internet users to give up their anonymity, privacy, and security before they access lawful speech." Verifying a person's age can be done without exposing their identity. It requires a trusted third party, such as the government, to issue age certificates. The internet company can then send the user a random challenge (just a random number), and the user can then prove that they possess a valid age certificate stating that they are over 21 without disclosing the certificate, their identity, or their actual age. This software technique is called a Zero-Knowledge Proof.
This is the wrong tool for the job. Zero knowledge proofs are simply a means for both parties to agree they possess the same secret information without divulging any secrets to an eavesdropper in the process. There always has to be some OOB means of both parties selecting which set of secrets they will be agreeing upon.
The technology that would be used looks more like an authorization token. The Internet site has a trust anchor comprised of public keys of age verification services. The user gets an authorization token from the verification service and hands it to the Internet site which uses the trust anchor to verify authenticity. The authorization token would simply be constructed in a way that does not divulge identity.
This would deny the Internet site access to the users identity while concurrently just as importantly never requiring any communication between the Internet site and age verification service itself.
The problem with this approach is the user would either keep having to get new tokens from the verification provider or such tokens themselves would end up effectively serving as tracking cookies. I think it could be largely solved by the provider belching out a bunch of different tokens at once the user could rotate between before getting more.
This leaves collusion with the age verification service which I can't think of a single reason anyone should ever trust. Even if they could the system would be completely worthless in practice given the effectively infinite pool of over age people identities could be lifted.
To say nothing of the obvious political issues with these schemes being fundamentally incompatible with a free society.