Blizzard accounts are Unique only in that they have a unique number appended to the end. For example there can be many 'eeeeeeeeeeeee' accounts, but only one 'eeeeeeeeeeeee#1432'. The games only show 'eeeeeeeeeeeee' during play.
I tend to like the approach Valve takes with Steam, which is to completely separate the "account name" (unique, unchangeable) from the "display name" (non-unique, change more or less as often as you want).
I have uh personal reasons for having done this and I know of others who have as well.
I’m not sure why they don’t roll it out more wisely - though I can say it did cause a couple of bugs until I signed out and back in on every steam device.
Each steam account does have atleast 3 internal unchangable account numbers that are exposed through APIs for developers to integrate with and use for things such as enforcing bans etc.