argentine is a noun, but in the UK we use Argentinean as the adjective. Example: Agentinean "star" Carlos Tevez at West Ham. Argentinean Warship the Belgrano etc. US and UK standard English have differing systems for things like that i think.
It is 'Argentine Tango' because it is called "Argentine tango" by us to differentiate it from ballroom Tango. In Argentina they don't call it Argentine Tango, they call it "Tango" because it doesn't need a qualifier.
What we call things from other countries comes from what we call the countries for example we don't call french things "francais" or German things "Deutsch". Like I said though it is a historical thing I know people who still say "Rhodesian". In the US you are almost entirely immigrant populations so it seems more natural that groups would refer to things in their original language out of habit, or find something as close as possible to it. But I would imagine if you spoke to an old Virginia family they would use the Anglicised version.
BTW has anyone heard Carlos Tevez's band?? I want to know if it would be entertaining to use his songs at a Milonga as a cortina or something. It's not Tango music as far as I know...
I live in TX and have never heard Argentinean used here in any of the cities I've gone to dance or have class...hmmm, intruiging. The only time I have ever heard it used was watching a British "Dancing with the Stars" type show. Mostly, I just say Tango, as anyone who knows me knows I don't mean ballroom. If I think I need to qualify it, then I say Argentine.