I'm having a similar problem only I haven't been locked out yet. But I am finding myself in an endless loop of trying to sign into an Apple ID account via the App Store on a new MacBook Pro for the school I work for. The Apple ID works on the website via a browser (like right now) but every time I try to sign into the App Store or iCloud on this machine, it says that I've never signed in this way before and would I like to review my information? If I click continue, I have to immediately sign in again (each time triggering a verification code, and each time I get a message that pops up telling me I haven't signed in this way before and would I like to review my information.
Unending.
Very annoying.
If this is some kind of limitation because the school doesn't have a credit card associated with this Apple ID yet, then the darn message should just say so. As it is, it just appears that I cannot sign into an Apple ID account that does indeed work on the apple.com website.
What gives?