The analogous action is to only require age-restricted sites (or parts of sites) to check ID, not the entire Internet. e.g. no one is calling for mathisfun.com to check ID. I'd expect most parts of the web are child-friendly and would not be affected. Just like how almost all locations in physical space don't need to check ID.
Additionally, the laws I've read mandate that no data be retained, so you have stronger legal protections than typical credit card use, or even giving your ID to a store clerk for age restricted purchases (many stores will scan it without asking, and in some states scanning is required).