Hard to canonically answer, but as long as you haven’t made the inputs and error message inaccessible to assistive technology, the only issue I foresee is what happens when someone tries to reset a password or needs to access any other system-generated email. With fake email addresses, the users will never be able to receive a password reset email and thus will have to have another way to reset the password.
Perhaps you could use some other method such as accepting their real email address upon registration, but putting them in an approval queue. An admin (or maybe it would be possible to automate?) would then set up an email address for that user that would just include the user ID, and that email address on the site’s domain would be set up to forward to their actual email address. Once the new email address was set up, WP would have no record of their actual email address, so you’d have that info briefly but never for long periods of time.