When people say sitting "Indian style" they evoke something from the 20th century. In Europe the position of sitting cross-legged on the floor was called sitting "tailor style" or tailor fashion, and Turkish style by the Polish and Romanians. Of course these are all colloquialisms.

So why "Indian style?" Well, the first account comes from a 1901 account of a meeting with Sioux natives:

They paid no attention to me, but seated them- selves, Indian style, on the ground in a circle in front of me

Though the term was likely in use some time before that.