You say there is no question, but it is already being raised by social media companies as an issue with the law. Anonymous age _verification_ is not something that exists. This requires government documentation, which means any solution that involves storing identity verification information would de facto allow the government to unmask users. This raises serious free speech concerns for everyone that uses these websites in Florida.
A fairly basic version of anonymous verification would be that local stores sell an age token. You show them your ID, like you would for buying alcohol, and they give you the token which could then be used with online services to prove you're over a certain age.
It only becomes a problem if stores are forced to store the ID and link it to the token in some way.
1) 3rd party service (Verifier) receives government documentation proving that the owner of {{account ID}} is over the age of 21.
2) Verifier records the fact that the owner of {{account ID}} has proven they are of the age of majority. They throw away all other identifying information, including their exact age.
3) The individual logs in to Verifier from Social Media. Verifier certifies to Social Media that the individual is over the age of majority.
There's no technical barrier preventing this outcome, and if you read the text of the law this is very clearly what the legislature envisions. Will it work? I don't know. Let's wait and see.
Let's just ignore the fact this doesn't exist for the sake of going through the mental exercise. Whether identity services are allowed to discard the logs is a legal question. The time in which a user makes a verification request for a website is de-anonymizing in of itself. The government could access both services either in real-time (remember this has already happened here) or depending on how things are stored at a later date to link identities to accounts. So the government has ways get all the information it needs to de-anonymize people who use social media in Florida if enforced.
You skipped over the Verifier expecting to be paid in some manner, being very likely to be paid using credit card, and thus having to retain those logs. Thus still making it easy to unmask users.