Hey everyone!
So I recently had this problem when I upgraded from 7 to the X, and it has been bothering me for weeks. After logging into my iCloud, messing with my settings and what not. I finally figured out how to turn off that annoying notification of approving the iPhone.
First off, on your device, you go to the Settings and just underneath your Apple ID, there's a random tab that is NEVER there, and it's saying you need to approve the device. Click it, and run it's course. Once it pops up saying you have to approve it on another iCloud device, in the bottom left hand corner, there should be some text there (for some odd reason I couldn't read what it said because it was cut off), click on it. It says you cannot confirm the device, and it resets the iCloud keychain feature. Then poof, the notification goes away. Next time you go into your settings, do NOT, I repeat, DO NOT, click on the keychain feature. With that feature, you have to have 2 devices (iPad and iPhone, or 2 iPhones) to complete that feature, all signed into the same Apple ID (I'm assuming, I could be wrong). But it does take that pesky notification off.
I truly hope it helps others like it did me. I never experienced that before and just fiddling around with it, I was able to turn it off. I hope y'all have a wonderful year!