I'm not sure what has changed to warrant all the hoopla. When I moved from Ohio in 1995, I went to get my Oklahoma license and was told that my Ohio license was not enough; they needed to see my passport and visa. I had to go back home and dig them out. That was not an isolated incident. Four years ago, when it came time to renew my license, I went to the tag agency to get it renewed. "Are you a U.S. citizen?," I was asked and when I said I wasn't, I was told that I could get my license renewed only at the DMV. That time, I had to show my green card at the DMV.
One clue may be in this article that ran in a local paper, where someone is quoted as saying:
"I guess what bothers me is there was nothing out that warned a person about this. We thought the bill was all about immigration, we didn't think it applied to us."
"Didn't apply to us" indeed. Perhaps what has changed is that the law used to be discriminatory -- if the person looks like a foreigner, ask for proof of legal status, but if they look like they belong, no problems. And now, perhaps, the discretion (and the potential to do ethnical profiling) has been removed. Just guessing, though.
If you know what exactly is new about this law, let me know. Because in my experience at least, you have always had to show proof of legal status before getting a driver's license.