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." 

Note that, even though the latest iOS, iPadOS, or macOS devices are mentioned above, sometimes two-factor authentication is not available. "While most iCloud users can now turn on two-factor authentication, certain account types may still be ineligible at Apple’s discretion." See Availability of two-factor authentication for Apple ID https://support.apple.com/en-us/HT205075

__________

But, note that Apple server maintenance may be what is causing it your problem. A few days ago I saw a post indicating that, per an Apple tech, Apple is conducting major [server] maintenance and that many services are down. The instruction given was to "try again tomorrow ... [and] maybe the [needed] server will be back online.

Not all accounts are eligible for 2 Factor Authentication. I suggest you contact Apple ID Support for assistance . 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