It can't be anonymous if you need a google or apple account to use. I'm not concerned about what the government tracks (well, not in this context at least), I'm concerned about who and what they're working with to do the tracking. If the app verifies me as an adult but I couldn't use the app without google butting in, google now has yet another data point in a secret ad profile that the government should be putting a stop to, not helping them build up. It'd be like announcing a plan to stop illegal drug usage by partnering with the cartel.
If they wanted a government-sponsored age verification sort of thing, it should've been an app whose only job was to type in a code you got from going in person to some government body and verifying in person. Town office, DMV, somewhere like that.
More fundamentally, though, "protecting the children" shouldn't go anywhere near anything that can be used for identity theft. Showing my ID to the cashier at the cigarette shop is significantly safer than showing it to any business on the internet, because sharing a high-quality picture of something is giving them a copy. The cashier gets to see it, but it never leaves my sight and isn't recorded in any way except probably some dodgy security camera where you can't read it anyway.