If the cert is not unique to a person, it's not a viable way of verifying age. Anyone could use one issued to anyone else, and the whole thing is DOA.
So then we have to assume it is uniquely tied to a person. Then, it can be used by the verification service to track a person, building a profile of what websites they visit and when, even what they do there, depending on how the age verification is set up.
Advertisers don't really care about your name, just about your profile so they can target you with ads, so it gives them everything they want.