I have an interesting twist on this error!
I received three emails from Apple - one that my billing address was changed, one that my date of birth and security questions were changed, and the last one that a purchase was made from a previously unassociated device in China. It seems pretty clear that my account was hacked.
No problem, I perform the reset password via e-mail option and change the password to something more secure. Next I try to log into the account to reset everything they had changed. In order to log into the account, it wants my security questions - the same questions that were just changed or setup by the hacker. (My account is pretty old, so I probably didn't have any security questions on the account since Apple added them in 2012 according to Google.) When I try to reset the security questions, I get the above error ("We do not have sufficient information to reset your security questions".)
No problem, I'll just call Apple. I call Apple and explain the situation. They need two pieces of information: a security pin that can be generated from the login screen (no problem), and the answer to one of the security questions. "Uhhh, remember when I just told you that the security questions were changed by someone in China?" Doesn't matter, they need one of those answered to unlock the account for me. They then coax me to try and remember the security questions, even though they were not questions I would ever choose. Then my account gets locked out for 8 hours from trying too many security question answers. The best part it that it also locks the Apple care specialist out of the account (what system doesn't allow customer service to override a lock???) So now I have to call back in another 8 hours. I also asked them what would happen if we can't guess any of the security questions. Their answer: "we would have to delete the account." It doesn't seem to matter that I have access to the account e-mail address, or 10 years of purchase information for the account, or that I have a string of e-mails that they sent me that clearly shows someone in China just broke into the account!
Luckily with the exception of a few songs I purchased like 10 years ago, I only use the account to buy itunes gift cards for the family each year. I would be furious if I had lots of music and app purchases like some people do. So 8 hours from now I'll see if the account can be unlocked, otherwise I'll have them delete it and start giving out Amazon gift cards for birthdays instead.