This was part of the red flag rule a few years ago. Currently, our policy is just to show a picture ID, to verify identity. We cannot make a photocopy, and our EHR allows us to scan in the license as a form of photo ID, but we have to crop the photo to only show the face, but no other information. If patients refuse to show a photo ID, though we don't refuse to treat them.
In Dec 2010, President Obama signed a Senate bill that excluded physicians from this particular rule. (see link below). The purpose of the red flag rule had to do with identity theft with creditors, and in a legal loophole, physicians were considered "creditors"...which of course we are, since virtually nobody pays cash for medical services these days.
http://www.healthcarefinancenews.com/news/obama-signs-law-excluding-doctors-red-flags-rule However, in light of healthcare privacy laws, the senate saw reason, and excluded physicians from the rule.
I would make sure you have an office policy, ask for the photo ID to verify that they're who they say they are (so they aren't trying to pass off someone else's insurance card as their own), but I probably wouldn't photocopy license or SS# information for a paper chart. Not only are you not required to, it's probably not a good idea to do so.