The California law doesn't strip away any anonymity or privacy except for the additional fingerprint signal of you being a kid or not, which is no worse than Accept-Language
It kind of does. Depending on how the mechanism works, if I check the user's age every time my executable is launched, and my user launches daily, I can determine with certainty what the user's birthday is. That information may be enough to deanonyimize in rural regions. It certainly gives away pii.
One allowed way to implement it is that your mom has to log in as root and unrestrict your account, and the user's age is only set to one of 4 different brackets so you'd better be watching the user every day for several years. Also if you are root you can just lie.
That additional "fingerprint" makes it easier to track people online. If the system was meant to protect privacy and anonymity, the signal would only be present if the user was underage.
