Duke, for example, is a Methodist University. As is Emory and a bunch of others

They're not really actively religiously involved, but Brandeis is pretty actively Jewish and is AAU. The Catholic University of America was AAU until 2002 so if there's a ban on religious schools, it's relatively recent

I think it has to do more with the focus on research and possibly the values of the school (which obviously have something to do with religion, but aren't always as driven by that as they are at BYU)