Mac keeps popping up Approve message "This device wants to use your iCloud account"

Welcome -Raspberry-,

Thanks for participating in the Apple Support Communities, and congratulations on your new MacBook!

Based on your description of what you're experiencing, this sounds like it may be related to iCloud Keychain. Let's check to see if that is set up on your Mac. Try enabling this on the other to see if that will clear this up. You can find more information regarding iCloud Keychain here: Set up iCloud Keychain

Turn on iCloud Keychain on your Mac

1. Choose Apple menu  > System Preferences. 

2. Click Apple ID, then click iCloud in the sidebar.

3. Select Keychain.*

4. If you choose to "Approve Later" when signing in to your Apple ID, you need to approve with an old passcode or from another device when prompted. If you're unable to approve, reset your end-to-end encrypted data when prompted.

*On your iPhone, iPad, or iPod touch with iOS 13 or later, or on your Mac with macOS Catalina or later, two-factor authentication is required to turn on iCloud Keychain. If you haven't set it up yet, you're prompted to update to two-factor authentication.

If that isn't helpful in clearing this up, Contact Apple Support directly to check into this in detail with you.

Have a great day!