Re" iCloud won’t authenticate on new Macbook When I try to log onto iCloud it says authentication failed on both my account and my son’s account - it’s for him and I’m setting it up with me as admin user and him as a standard.
The Apple Account is the account you use to access Apple services like the App Store, Apple Music, iCloud, iMessage, FaceTime, and more. It includes the email address and password you use to sign in as well as the contact, payment, and security details you use across Apple services.
Check whether you have correctly signed your son (or yourself?) into the new MacBook with the correct Apple Account details for using iCloud, FaceTime, Messages etc..
See how to: Sign in to your Apple Account on the web or in your device settings - Apple Support
If needed: Using the new Mac: Sign in with the correct Apple Account details online to ensure the new Mac gets listed there as a trusted device: https://account.apple.com/
How to: Get a verification code and sign in with two-factor authentication - Apple Support
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More info: Use two-factor authentication for Apple Account security on your Mac - Apple Support