No. Reciprocal state issues do not become federal issues. My California drivers license is accepted by law enforcement in all 50 states. It does not become essentially a “federal” drivers license. It is still a state drivers license. It is just accepted in the true meaning of “federalism” that has basically fallen by the wayside due to the over-reach of the US central government.
The reason your license is accepted in all other states is that if is effectively federalized.
During a Jim Crow, a mixed race couple could be married on one state but another state didn’t have to recognize it.