It can be done. The website provider can generate a request that it forwards to you. You then pass on this request to the age verifier, who can answer "yes person is over 16" without knowing why you want to know, or who generated the request.

The requester wouldn't know your age, just that you were old enough.

There are a few problems.

One is that the website could embed some identifier in the signature of their request. But any information there can be easily send by the web site provider to the age verifier directly if they wanted so this is not a big problem.

Another problem is that the age verifier could look at times when requests were submitted and create a sort of "fingerprint" based on when requests arrived for different sites. This could be partially helped by having browsers request age verification randomly in the background any time you use a browser.