They cannot take you into custody and take you down to the station without probable cause. Not having an ID card is not probable cause.

In most states, (CA and CO are exception since traffic violations are criminal offenses) cops cannot take you to the station for a traffic violation if you provided your correct name and signed the ticket.

Even if cops have reasonable suspicion for a "Terry Stop" they have only a limited amount of time to conduct the investigation which might give them probable cause. The Supreme Court has not nailed down a time limit, determining that on a case by case basis, BUT it seems that 45 minutes is about the most they are willing to tolerate the cops detaining a person for most cases.