Hey there EJLC94,
Thanks so much for using Apple Support Communities.
This definitely sounds like unexpected behavior. When you go to log in, you should see a screen like the one seen below:

Are you not seeing that box that asks for the verification code? This screenshot is an iPad, but the verification code should look the same.
If you're not seeing that at all, we would recommend that you restart your Mac and test the issue again. Sometimes a simple restart can get things working again, so it's a good step to try. Once you've restarted, if the issue continues to occur then we would recommend seeing if you can log in to the iCloud website or the Apple ID website, just to try and isolate the issue to one specific app or process that isn't working as expected.
Here's where you can try logging in to test the issue:
iCloud
Manage your Apple ID
We hope that this information is helpful.
Have a great day!