I have read only few paragraphs and not sure for details of this law but it sounds like OS requirement, not browser's:My reading of it is that the (admittedly highly ambiguous) description of the mechanisms given in the legislation can be entirely implemented in the browser and target websites, without requiring any change to the underlying operating system on the device being used to access those websites.
"The OS must show an interface during account setup. This screen requires the user to provide their birth date or age."
If the responsibility is for OS manufacturer (developer) plus open source OS, it looks like impossible mission. 13yo child can get the OS in source form, remove 5-10 lines, compile and have unrestricted OS.