The civil rights of citizens are the purview of government, elected by the citizenry. Religious organizations are not people. Most of them, certainly the worst, most powerful and most dangerous, are little more than corporations. A religious organization that motivates and organizes public action for the homeless is no different from an organization that organizes for the right to keep and bear arms like the NRA (a Tea Party favorite).
The big difference is that most advocacy groups weren't founded on a sort of ideology that ripped cultures to pieces with sectarian bloodbaths for thousands of years. Religion deserves special attention in that regard.