I am coming late to this topic, but I think it is a very rich and complex one. Offhand I'd say that what makes Swan Lake in particular "Russian" is the fact that Tchaikovsky wrote the music. That makes it "ours" from a Russian perspective in a way that Giselle could not quite be, for example, despite its restaging by the (French) Petipa in St. Petersburg in the 1860s.
In a larger sense, though, I think that what makes ballets Russian is the fact that they are "archived" by living Russian institutions, the Maryinsky, the Bolshoi, and others. Unlike a painting, which simply hangs on a wall and is viewed, Swan Lake has lived and passed on through Russian dancers and teachers. It survived the Revolution - with a changed, more uplifting ending - and continued on through the Soviet era, passed from dancer to dancer. That direct physical link, along with the workings and reworkings of this ballet (Gorsky, Vaganova, Lopukhov, Sergeyev), make it Russian, even if there are no overtly Russian dances in the ballet (the Russian dance not being performed in contemporary productions). And there is a lot of consciousness about Swan Lake being a sacrosanct tradition in Russia, even if the ballet went through a lot of changes. I don't think you see that in the States. Swan Lake is just another ballet to be performed, albeit one with its own special style and history, but not one to be approached with reverence and awe.