The truth is that Franco had no club, he hated football.
He made a bit of a show of clutching on to Madrid because they were so successful internationally at a time when Spain's image (and society) was in the gutter, but that's all it was, a show. He did the same with the singer and actress Lola Flores (Quique Sánchez Flores's grandmother incidentally) who was similarly famous abroad. She was a gitana and Franco hated gypsies, but like Madrid, he thought she gave Spain some international credibility.
But you're right to a large extent, Atlético de Madrid were previously known as Atlético Aviación, the Air Force team and continued to be seen as the "military's club" for some time after. And about their fans too, Frente Atletico are very much are still a thing, long after the Ultra Surs at Madrid were shut down at the start of Florentino's presidency
