iOS 10 stuck on approve this iPhone from another one of your devices
This is almost certainly a keychain issue. Try these steps:


Turn off iCloud Keychain on both the MacBook and iPhone.
From your iPhone, go to Keychain and tap Approve with Security Code.
When prompted for the 4-digit iCloud Security code tap Forgot Code.
Follow the prompts to reset the code.
Now this device should have iCloud Keychain enabled and working.
Now from your MacBook, enable iCloud Keychain.
You should get a notification on the iPhone about this. Enter your iCloud password in the prompt.
iCloud Keychain should now be enabled.


Hopefully this gets rid of the Approve your device notifications! Let me know how you go.

[EDIT]
***Make sure you have a passcode setup on your iCloud Keychain on your computer. If you don't set one up and then try approving your phone again. That is what made mine work. Hope this helps!