Of course you can. All you need is a trusted 3rd party (the organization who issued your ID, probably your government) to verify your identity and sign a statement that you are over 16 years old. Then you present that statement to the social media company and you’ve done it.
By using an identifier like a session token the verifier never has to know what website you’re visiting. You show the social media company a message containing this season token, an assertion that you are over 16 years old and a signature. You don’t need it to delete or securely process that data, as the only thing it knows is that you’re over 16 and it’s required to verify this for all users.