This law doesn't do anything that prevents non-anonymous access. Here's how you would access things anonymously if you bought a new computer that implemented this.
1. When you set up your account and it asks for your birthdate, make up any date you want that is at least far enough in the past to indicate an age older that what any site you might use that checks age requires.
2. Access things the way you've always done. All that has changed is that things that care about age checks find out you claim to be old enough.
The only people it actually materially affects on your new computer are people who cannot set up their own accounts, such as children if you have set up permissions so they have to get you to make their accounts.
Then if you want you can enter a birthdate that gives an age that says non-adult, so sites that check age will block them.
From a privacy and anonymity perspective this is essentially equivalent to sites that ask "Are you 18+?" and let you in if you click "yes" and block you if you click "no". It is just doing the asking locally and caching the result.
