Third party, perhaps. You sign up for service A. Service A queries service B, which knows who you are and provides a one-time ack of your age.
It's virtually impossible to make a verification system that's anonymous. Somewhere the third party and authenticator will need to share a secret that you cannot touch.
Furthermore, you would need the government to agree to this system and mandate this system universally and pay for the authentication services to exist. That's not what Florida is doing.
The problem is that my ID contains too much information; I would prefer a document (i.e. digital certificate) that only certifies my age, not my name, address etc.
You cannot get around this. The service must confirm with the authenticator. The authenticator must know you are authenticated, and be extension, who you are.