Unable to Sign-in to Apple ID account

I have received a second hand laptop in work (had been used by a pervious colleague) and I am trying to change from my old MacBook to this new MacBook.

My colleagues account has been removed.

I have my own login user which I have logged in to, but I can't sign in to my Apple ID account

I am trying to sign in to my Apple ID so that I can download from iCloud, but the login is not working.

1. I enter my email address.
2. I enter my password
3. NO Verification Code is sent to my older laptop.
4. I select to send a code to my phone, which I get.
5. I enter the code and the login processes, but refreshes with no account signed in to Apple ID.
6. On my older MacBook I can see the new MacBook linked to my account, but on the new MacBook the account is not signed in so I can not download my iCloud data.
7. I tried logging out of my account on my older MacBook and starting the process over (after removing the new linked MacBook) and in this case the verification code and new sign in pop up DID show on the older MacBook, but on the newer one the account STILL did not actually log in.
8. I actually created this question on the newer MacBook and when logging in I had to request a verification code, which did appear on my older MacBook (while logged in) and everything worked.... but it simply does work to actually connect my new MacBook.

I can't understand why the login process appears to complete, no error is shown, but the login doesn't actually complete (despite showing as linked to my AppleId)

[Re-Titled by Moderator]