The bill specifically states forms of identification, not licenses to drive. I'm not sure identification would fit the definition of full faith and credit, since every state gets to decide what identification means and is required for.
Driver's licenses are a different story. Those would be respected if valid in another state.
Full Faith and Credit shall be given in each State to the public Acts, Records, and judicial Proceedings of every other State. And the Congress may by general Laws prescribe the Manner in which such Acts, Records and Proceedings shall be proved, and the Effect thereof.