This is the most annoying and inane process that Apple has failed to fully address! To fix this issue you just hit cancel, as the "Two-Factor-Authentication" is actually completed/approved! I spent hours linking device after device and following step-by-step instructions in detail until I wasted an additional 25 minutes sitting on hold over the phone while Apple "Geniuses" did their best to try and read from their support cards/help-sheets for better instructions. Their first answer was to hit the bottom link that is titled (Can't Approve This Device?) DO NOT CLICK and/or FOLLOW THIS PROMPT! It'll prompt you to follow steps of deleting encryption on device that will possibly erase iCloud stored photos, files, contacts, etc.
I refused, and stated this made absolutely no sense and I did not feel comfortable doing this at all! And that in doing so, it completely defeated the process of two factor all together as the soul purpose is an extra security feature to protect your info and/or help make third party Apps run correctly on your device such as G-Suite by Gmail etc. If I wanted to erase and not protect my personal photos etc, I wouldn't back them up to iCloud in the first place!
Finally the "Genius" came back with a great suggestion "that made total logical sense." Just hit cancel.' That it was in fact "already approved!" The "Genius" stated once you're done signing in to iCloud/following all steps it's done approving. Just ingnore the prompt and hit cancel.
Unfortunately for the majority of normal thinking individuals, following all steps including the "final" waiting/loading icon that appears for all of us in a never ending loading cycle; prompting us to "APPROVE Two Factor" is very stressful and doesn't make sense why it would do this if it's in fact already done! However we're just suppose to apparently assume that it is in fact actually done and just to hit cancel.
Reason I was given is that the stuck in limbo icon/process is due to (Keychain,) not TWO FACTOR, even though it's labeled ( TWO FACTOR). Also that it's a known issue that they have just failed to fix even though thousands of people if not millions have been extremely inconvenienced. "No Big deal!"
Long story short, hitting cancel to top left of icon box closes it, and it is setup. After doing so, I would recommend going into your iPhone and/or iPad settings and clicking at the top where your name/iCloud/Apple ID settings are, click
Password & Security. Then click under Two-Factor where it says Trusted Phone Numbers and add a secondary trusted number, to your already added number. This will help you down the road if you ever need to receive passcodes/verification codes but are actually unable to access your phone/device. Avoid this headache as much as possible!
Best of luck!
-iRie Vibes Odin