>No, this is an absolutely terrible idea. You're suggesting a giant, centralized, government-run data silo, with all of your online activity tied to your real-world ID. This is far worse for privacy than any data broker, it's hard to even compare.
Not all your online activity, even if they kept logs it would be something like 'this site asked for age verification, we said yes'.
So they would have a list of sites, if they stored them and were allowed to store them. Which is something they can get from your ISP regardless.
It could be used for bad sure, lots of things can. In my perfect world this wouldn't exist at all like it hasn't for 30+ years. But putting the burden on private companies was always going to create other avenues for issues.