The Sari actually predates the Greeks by quite a bit, with the earliest representations of them in Indus Valley artifacts. There are also references to it in the Vedas.

I think the similarities though are basically due to the simplicity. A sari is basically just a large strip of cloth that is wrapped around the body. When weaving was just coming along, a soft rectangular piece of cloth would be the easiest thing to make and it would just naturally follow that multiple civilizations (including the Greeks and the Indians) would use it as covering. With little need for additional insulation, this style would be functional, simple, and extremely utilitarian and so was quickly adopted and spread.