Two Factor Authentication

Just worked through this issue with Apple support and in my case it turned out to be an AppleTV unit that was linked to my iCloud account. If you're still experiencing this issue, go to iCloud in your device settings to see if any of the devices linked to your account are older or if there is an AppleTV linked to your account. Remove any old/unused/unnecessary devices from your iCloud account, restart your device, and try to activate two-factor authentication again. This solved the issue for me, hope it helps others.

Make. sure you can use two-factor authentication.

See Two-factor authentication for Apple ID https://support.apple.com/en-us/HT204915  See this section especially Setup two-factor authentication for your Apple ID. From this section, "Two-factor authentication is available for Apple ID accounts with at least one device that's using the latest iOS, iPadOS, or macOS. Learn more. You can follow these steps on your iPhone, iPad, or iPod touch to turn on two-factor authentication."

And, Use the “learn more” link above to see Availability of two-factor authentication for Apple ID https://support.apple.com/en-us/HT205075

Certain types of accounts, besides those listed above, can't use 2 Factor Authentication. Contact Apple ID Support for assistance/information. See the Apple Support articles below for contact information.  

Apple ID - Official Apple Support.    

Contact Apple for help with Apple ID account security - Apple Support.   (Link doesn’t always load)  

Get Support          Work through the screens to get a chat option. 

Apple Support schedule phone call

https://twitter.com/AppleSupport