My DS won his congressional slate and has a direct nomination to USMA, but is also applying to USNA
Perhaps USMA has different terminology ...but mixing up terms makes this a confusing statement for USNA. First, did Candidate get a nomination . Most nominations are "competitive", with the MOC nominating up to 10 candidates. Then, USNA Noms and Appointments chooses the most competitive from among those 10 candidates and offers the Appointment. "Winning the congressional slate" refers to being awarded the Appointment from among the nominations, not receiving a nomination.
As to the orignal question -- Nominations to each Service Academy are independent of each other. A candidates can get a nomination to multiple Service Academies, although many MOC try to spread the cheer by apportioning them according to the candidate preference.