Email addresses aren't real-life identifiers, although they could be used that way. And on the inverse, I can buy a burner phone with cash and defeat Blizzard's requirement easily.
But it's pretty clear what Blizzard is going for here, since we know their platform is competitive and thus harsher on individuals viewed as bad actors. Blizzard knows full well that most people signing up to play only have one phone number, which is most certainly a real-life identifier for those players.
My point is that if players feel compelled to self-censor their interactions online for fear of repercussions[0], then they are not playing a game. They are instead representing themselves as individuals on a social network with competitive/gamified attributes. Requiring phone numbers to play is just a method to gently reinforce this.
To each their own - this might be exactly the type of game you like to play. But it departs pretty strongly from the classical definition of "video game" that I grew up with.
[0] For example, your primary phone number being disallowed access to all Blizzard services. Maybe they go even further and request that 3rd party services ban you for off-platform violations. Moved this down to a footnote so as not to spread FUD, I'm sure they don't currently do this, nor do I expect them to. But cross-referencing phone numbers would make it very easy to start.