So this issue have been here for more than a year if not more, I so feel for all of you have been so frustrated by this as I was too, I have been staring at the phone and computer for more than 8 hours. By LUCK, I was able to "approve this iPhone from other device". Here is what happened during this process: I did not see "approve" button anywhere for three hours, tried to log in to iCloud as it asked, as you all said/knew by now, there is no "approve" button anywhere while you are in iCloud. Absolutely FRUSTRATING. As OCD as I am , I just can't stop trying again again and again. after many times, I have no idea how the approve screen popped up on my other device(another phone that is signed in to the same appple ID) like one of the guys here said ( suggest to go to do something else for a little while and come back) and maybe it would happen. After the approve window popped up on my screen on my old phone, I typed in the apple ID password and clicked " Approve", guess what happened? NOTHING HAPPENED! here I'm again trying type in the password, and clicked "back" arrow on the new phone I was trying to set up - the one stuck on the " Approve this iPhone from other device" , so it would start the "setting up apple ID" process over. I tried about 7 or 8 times ( after I input the password , nothing happened). Then I saw my old phone is in an older version since I haven't been using it for a while, I decided to update the phone, hope it would help. And it probably did, because on my first try after updating the old phone, the "approve" window popped up again, it worked.
I'm so sorry this have to be the way we resolve this. Really- Apple? !!
I hope I explained myself clearly. it's 4:20Am here, after 8 hours trying, I may not be in a clear mind to describe all this :)
Reviews have helped me so many times before, so I decide to write one in hopes to help somebody, at lease you know many people had trouble with this issue.