Do people in the US military who didn’t go to one of the academies care about the game or is it mostly just a student/alumni thing?
Mostly the latter. I had a childhood friend who’s a West Point grad. He told me that in Pentagon staff meetings in the weeks before the game, the last thing said at Army briefings is “Beat Navy!”
Speaking for myself, I didn’t go to either place and it doesn’t matter to me who wins the game. But the original lyrics to “Anchors Aweigh” contain the lines “sink the Army gray”, so other sets are merely an adaptation.