It's bittersweet. The National Cemetery system was set up at the end of the Civil War, and it was decided that the dead from both sides of that conflict would be honored by burial in those cemeteries. In the case of unknown soldiers of both sides, they are typically not identified as Union or Confederate.
Lincoln's instructions to Grant before the surrender was to "let them up easy." There are, of course the dead from all wars since, and veterans are eligible to be buried here, if they so wish.