Cannot...Finish setting up two-factor authentication on this Mac

You could try a number of things to try and fix the problem.

A Troubleshooting Procedure that may Fix Problems with macOS El Capitan or Later

You should try each, one at a time, then test to see if the problem is fixed before going on to the next.

    Be sure to backup your files before proceeding if possible.

1. Shutdown the computer, wait 30 seconds, restart the computer.
2. Disconnect all third-party peripherals.
3. Resetting your Mac's PRAM and NVRAM
4. Reset the System Management Controller (SMC)
5. Reset your Startup Disk and Sound preferences.
6. Start the computer in Safe Mode. Test in safe mode to see if the problem persists, then restart normally.
7. Use Apple Hardware Test to see if there is any hardware malfunction.
8. Repair the disk by booting from the Recovery HD. Immediately after the chime hold down the Command and R keys until the Utility Menu appears. Choose Disk Utility and click on the Continue button. Select the indented (usually, Macintosh HD) volume entry from the side list.  Click on the First Aid button in the toolbar. Wait for the Done button to appear. Quit Disk Utility and return to the Utility Menu. Restart the computer from the Apple Menu.
9. Repair permissions on the Home folder: Resolve issues caused by changing the permissions of items in your home folder.
10. Create a New User Account Open Users & Groups preferences. Click on the lock icon and enter your Admin password when prompted. On the left under Current User click on the Add [+] button under Login Options. Setup a new Admin user account. Upon completion log out of your current account then log into the new account. If your problems cease, then consider switching to the new account and transferring your files to it - Transferring files from one User Account to another.
11. Download and install the OS X El Capitan 10.11.6 Combo Update or 10.12.6 Combo Update or Download macOS High Sierra 10.13.5 Combo Update or Get MacOS Mojave now from the Mac App Store as needed.
12. Reinstall OS X by booting from the Recovery HD using the Command and R keys. When the Utility Menu appears select Reinstall OS X then click on the Continue button.
13. Erase and Install OS X Restart the computer. Immediately after the chime hold down the Command and R keys until the Apple logo appears. When the Utility Menu appears:

     1. Select Disk Utility from the Utility Menu and click on the Continue button.

     2. When Disk Utility loads select the drive (out-dented entry) from the Device list.

     3. Click on the Erase icon in Disk Utility's toolbar. A panel will drop down.

     4. Set the Format type to APFS (for SSDs only) or Mac OS Extended (Journaled.)

     5. Click on the Apply button, then wait for the Done button to activate and click on it.

     6. Quit Disk Utility and return to the Utility Menu.

     7. Select Reinstall OS X and click on the Continue button.

Continued from above:

14. If none of the above helps then see How to Downgrade macOS High Sierra and macOS Reversion- 

      How to Downgrade from High Sierra.

15. If you get here without success then it's time to make an appointment at the Apple Genius Bar for 

      service. If you need to find an Apple Store - Find a Store - Apple.

Two-factor authentication for Apple ID - Apple Support

Apple ID - Get a verification code and sign in with two-factor authentication

Still unable to turn on two-factor authentication even when the Set-up button is present

If you enable two factor authentication, keep in mind that cannot disable it anymore.

In the system preferences > iCloud if a notification comes to set up 2FA with a yellow exclamation mark .Click on continue .

It's your choice to set up 2FA by clicking on turn on and click on continue , the 2FA will set up but that can be converted once again into secondary authentication before fourteen days a method is there ( using an iOS device and a Mac simultaneously ) , and if you click on not now the 2FA will not be set up .