IIRC, "Schatzi" was the name Adolf Hitler and Eva Braun gave their pet German shepherd.
The reference to Benito Mussolini's chin-tilting was cute, too -- although I think Obama is probably imitating another Latinate hero of his, Fidel Castro, instead. The resemblance doesn't strike most people right away because of Fidel's beard.
I think it's just a common (South?) German term of affection. Comes from "Schatz," = "treasure".