Yes they can. That is the problem with the vehicle laws and the courts. Its definitely not fair but a conviction puts points on your license. The Louden County (VA) Sheriff recently waged war on cyclists, and started giving tickets to cyclists for not stopping at a stop sign during a charity ride. A couple weeks later he called the local media down to announce he was starting a ticket writing campaign against cyclists. The local newscast quoted an attorney representing a few of these cyclists that they got points on their license.

So someone who does not own a car or have a drivers license is penalized less than someone who does, even though a license was not required to ride the bike, nor was a motor vehicle used for the infraction.

I agree its probably best to produce a credit card and some work ID with a photo or something like that, and hope your drivers license number doesn't get recorded.