In the past every state honored the driver's licenses of every other state. The Feds did as well.
Now the Feds are saying that they will not honor the driver's licenses of some states, i.e. they do not grant certain states 'full faith and credit'.
Full faith and credit deals with state to state relationships. And, indeed the Federal government can throw some monkey wrenches even into that:
“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.”
Using a state driver’s license to get onto a military base would fall under “To make Rules for the Government and Regulation of the land and naval Forces”