I have a US battle.net account, but I am studying abroad. When I tried to add a phone number for Overwatch 2, I was having issues.
However, I realized how to fix the problems I was having, it seems to be some silly bug:
Whenever battle.net webpages were opening, I was directed to links like “eu.battle.net/blahblahblah”. When I tried entering my phone number, there were error messages popping up… BUT, by changing this to “us.battle.net/blahblahblah”, I was able to add my phone number and play Overwatch 2.
This seems like a rather silly issue, I’d suggest somebody try to fix this.