Yeah, something like that. But while your device can validate the cryptographic sig for the app, the site requesting proof of age can't, since it isn't running on the same device as the app. The best I can guess, the app could request verification from the state run site, and specify what information it wants (based on what the requestor site asked for). The state site could use a private key to encrypt the response and give it back. The app could use a piblic key the state makes available to decode and confirm that only the intended information is present. Then the app can pass that to the requestor, who can get the public key from the state site and decrypt the information. But, the gap there is how does the requestor know the app it is talking to hasn't been modified. I don’t think there is a way that it can. Only the device the app is on can verify that. And the requestor can't trust the device either.
Some Authentication that I remember has a component where the requestor would then talk to the state to confirm the info it got from the app was requested from the state by the same app the site is talking to. This prevents using someone elses response as your own. But in this case, that would tie the site to the request which means the state would have both peices of info, who and what site. So I don’t know what there solution here could be that wouldn't result in the same problem.