All aliens should be required to get a license. This is no different than any country requires of Americans who move there.
But illegal aliens are necessarily, by definition, in this country illegally. At a minimum that should mean that they file the necessary paperwork to be here legally under whatever status that might be, or they should be required to leave.
The last thing you do is give them more legal identification and privileges with regard to motor vehicles.
So while I agree that it's not the states' job to enforce immigration laws, that doesn't mean that they should do things to make it worse.
I think the world of Jeb, but he's taking the wrong stance here.