The text said it doesn't but honestly when readiing the bill, age verification is the most likely outcome for websites having to create a database of users to store their IDs/drivers licenses on their servers.
Mandatory or even incentivized age verification would require a separate act of Congress.
One of the things I like about KOSA is that it separates the verification question from the "once you know, then don't do X" question. A more nefarious bill designed to destroy anonymous speech on the internet would certainly conflate the two issues in a single statute.
As someone who doesn't want ANY mandatory age verification on the internet, I very much want to fight that fight on its own merits and not wrapped up with something like KOSA. Passing KOSA separately and without Age Verification requirements has the advantage of giving a stronger position to anonymous speech proponents when and if mandatory age verification statutes are proposed. Both in Congress and also, potentially, in court.