The Eidah Chareidis is a group made up of different factions. Litvishe like Briskers and old Yerushalmis, as well as Chasidish. Rav Moshe Shternbuch has a Shul named after the GRA ztvk”l. Obviously he is not Chasidish.
The Eidah Charedis goes back many years, to old Yerushalmis. It was not started by Satmar. Satmar was in Hungary then. They came later.