I don't know about American tango, but i can definitely say that it is neither international ballroom tango, nor tango argentino (although the choreography is remotely mimicking some movements seen in tango shows, a style called sometimes "tango fantasía", yet done using an entirely different technique here).
I suppose that this is simply an ecclectic piece of choreography, using elements from different genres mainly for effect (the movie character wants to impress the students) and as a framework for the acting. Of course, it is possible that an actual tango style like this exists somewhere.
The song is sung by Veronica Verdier and called "Así se baila el tango", which is actually a quote from a different, very famous traditional tango
(which you can find at http://www.carriechelsea.com/songs/asi.html with music to listen, the lyrics and a translation).