You can use a driver's license or your bank. Different methods to verify it to the app. After verification, the app won't save the info, just the knowledge that you're over 18.
the app also knows every site you visit (that requests your age)
Not much of a worry locally. Disabling history is probably included in this (sort of porn mode/incognito, same as your browser). The info isn't much use locally anyway
In practice that’s just your government (as that’s the most likely source of authority to verify your age) tracking what you’re doing on the web.
Government knows you've verified the app but they don't know what you're using the app for, it doesn't have phone home. Your app is giving the site a generated cryptographic key, the site compares that to see if it is a valid key and then accepts it. The key doesn't carry your personal info with it, so the central db doesn't know who is being asked about, just whether the key is valid (so someone is 18+)