Thanks, Eric.
Here is what everyone should know about 2FA when they can't get it done:
The error message given when attempting to activate the two-factor authentication can be misleading or wrong!
Very often, the problem is that the own Apple ID account setup is not complete and some some important recovery information is missing to Apple. This might be either:
the primary e-mail address has not yet been verified (go to System Preferences > Account Details > Contact, in order to check and - if necessary - click the "Verify" button next to the e-mail address and complete the process).
make sure that you have set up your 3 security check questions (see https://support.apple.com/en-us/HT201363 for instructions) you must have provided a trusted phone number (in System Preferences > Account Details > Security
Apple really needs to add more info to their support pages on 2FA.
[editorial]
This whole mess of requiring a "3rd party specific password" issued by Apple is security-overkill. The reason I did it is because the Contacts app on my iMac is buggy... it truncates long contact notes and often when editing a note it will erase previous entered data. So I bought something that works much better: BusyContacts. And it requires 2FA etc. What a PITA!
[/editorial]