The Earth is closer but I don't think there's any good argument to say it's actually a binary system. Notably the Earth-Moon barycenter stays inside the Earth, and the Sun-Jupiter barycenter doesn't stay inside the Sun. But barycenters are too easy to get outside of objects anyway.
As I understand it, one suggested "double planet" definition was that the barycenter would have to be outside both objects. Earth-Moon doesn't (quite) satisfy that condition, but for example, Pluto-Charon does.
The Earth-Moon system is certainly anomalous within our Solar System. There is no such equivalent among the three other rocky planets, and while there are large moons around the other planets, the planets themselves are far larger.