And what does that information contain within? It can literally be just "over 18" and "under 18"
To the web site the government server will probably give just that information, yes. Internally the government service who hands out this bit of data will be tied to your real identity. It can therefore easily keep track of which websites are requesting this innocuous, pseudonymous "over 18" or "under 18" bit of information on your behalf. It can then record all of these sites in the profile the government server keeps for you. And if you believe that a government would never record such data, then boy do I have a bridge to sell you.