You can pick any username you want. But the username is appended with a pound sign and a 4 digit number. (for example, my Battletag is minimaxir#1667)
Everyone wins the branding wars. Presumably, there aren't 10,000 people who want the same name.
Pretty sure I've seen someone with a 5-digit number after the hash.
In any case, you're right, it's a great system. In most Blizzard games, the name you want is over your head. In rare circumstances you could have >1 person with the same name in the same game, but even then it's not going to be confusing because they'll be playing different heroes, or be your opponent, etc. Much better than WoW where people wind up choosing names like Årthäs to get sort-of the name they want.
However, the real reason this is a problem in the non-gaming world is that URLs and domain names need to be unique, so Blizzard's system can't apply.