And how would that app verify you've given your actual age? Now you need to deanonymize yourself to that app and while random sites might not know who you are the app does and the app also knows every site you visit (that requests your age).
So no, that's no longer anonymous. 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. Is that really preferable to parents parenting their children?