I would guess that their jurisdiction was written into law because of the predominant role of telecommunications and media in the commercial satellite launch market.

My knowledge is more certain about the international situation - the US has jurisdiction over its own objects because the Outer Space Treaty gives each party to it the right (and responsibility) to regulate the objects its citizens launch into space. I'm actually curious why the US decided that it was their job to regulate satellites launched by India but owned and operated by US citizens and corporations, but India seems to agree (because they didn't try to do the licensing for these satellites).