If you were in the service for 5 months and have a military ID card does that qualify you as a veteran on veteran day?

It depends on your definition of veteran. If it means that you served in the military, then yes. Many offers and discounts apply to any former service member who can prove their service. Many, however, also require an 'Honorable Discharge', versus, say, Other than Honorable, or Medical.

However, to join a veterans' organization, such as the VFW, you need to be able to prove that you were actually in a war/conflict.

I, personally, wouldn't try to claim it unless I served at least a full tour of duty, unless I was discharged for war injuries.