Can CiviCRM provide a warning/error if a user submits a certain organization name as their current employer?
The exact behaviour you describe would be possible but will have to be custom development. It would be specific to your organization and website.

However I do think you will be able to achieve something like what you want with the CiviRules extension. You would be able to 'catch' the new contact as a trigger (individual is added) and use the field value comparison condition to check the value of the current employer  (or alternatively catch the new relationship Employer of). CiviRules would allow you to then create an activity for a team member, alerting him/her to the problem. It is not perfect, but sounds like an alternative without custom development.

You can find the CiviRules extension here: https://civicrm.org/extensions/civirules and documentation about CiviRules here: https://docs.civicrm.org/civirules/en/latest/