iCloud Account not eligible for Two-Factor Authentication (2FA)? No reason why. Told by Support my Apple ID may be too old.

I attempted to upgrade my security to Two-Factor Authentication, but was not able to verify my account after multiple attempts. Then received two errors (depending on Apple Device being used to try this) - either "Two Factor Authentication Is Not Available For This Apple ID At This Time" OR "The Apple ID xxxxxxxxx@xxxx.com is not eligible for Two Factor Authentication." In talking with Apple Support through a few echelons of the Support Team, no one could give me an answer as to why not. At the end, I was provided an article from Apple that says some iCloud accounts may not be eligible and the tech told me my Apple ID/iCloud account could be too old. Too old? Should being a longtime user/customer disqualify me from being eligible for this feature? How can I resolve this issue and upgrade to this feature without trashing my current Apple ID and creating a new one?