Done correctly, the government entity would be one that already knows your identity and age, and the website would not interact directly with that government entity. They would only interact with you, when you give them the thing you got from the government entity that certifies your age.
Again, if done correctly, the thing you give them is not the actual thing you got from the government but rather the result of running that through a transformation. The transformed token can still be recognized as certifying your age, but if given to the government entity they would not be able to tell which token they issued it is based on.
If you were a web admin, or governemnt official with decision power in your hands, would you really choose to implement it like that instead of a direct communication between site and government server?
Again, if done correctly, the thing you give them is not the actual thing you got from the government but rather the result of running that through a transformation. The transformed token can still be recognized as certifying your age, but if given to the government entity they would not be able to tell which token they issued it is based on.