The problem for a lot of people is iCloud account is also use for iTune, and if you have credit as like me, sign up for new ID is not an option, at least not until credit is use up.
Added: For anyone shared backup, like password or other data in iCloud drive, it will required same login account, which means all iOS and all OSX device need to be transition to new account.
This is clearly an Apple iCloud issue, it need to be call into Apple support and ask them to address it. Work around it will only let Apple think there is no issue any more.