You mean modern Blizzard does that.
Oldschool Battle.net had a 16 alphanumeric characters with underscore allowed and that was that. At least for Warcraft 3 it's been the case since 2002 for a very long time (until quite recently last year they allowed fancy symbols in usernames).
You also had to login at least once every 3 months or Blizzard purged your account.
UUIDs were also my first idea, but I have the feeling that sharing them (i.e. to invite a new friend) would be cumbersome. I wonder if a new system akin to what3words.com could help there.