I mean yes, but then you'd also have to shut down Gmail, live.com and all the dozens of free email hosters.
Better make prepaid SIM cards illegal too. Oh an letters can be mailed anomalously, better make that illegal too.
Some countries go further and require government ID to sign up for even stuff like social networks (South Korea)
You propose it as a ridiculous conclusion but these are things that democratically-elected governments do.
In some countries anonymous e-mail is illegal. If your IP comes from such country, Google will _always_ ask for phone number on that account without option to refuse. Otherwise they will ask you after completing a registration (there is a delay to let you get hooked on, but Google still asks eventually). Some IP addresses are whitelisted — if you already have a "trusted" Google account, attached to that IP, or if Google recognizes your address as "corporate" or if you are an American and live in "liberal" state or... Basically, there are pockets of world, where Google does not ask it's existing accounts for phone numbers. But those pockets are shrinking and eventually you will be expected to cough up your phone. Unless you have already given Google your phone (e.g. by owning an Android device), which may create false impression, that you didn't need to.