I think either word could work

But somehow you don't consider being "anti-Islam" (read: anti-Muslim, because people with that set of prejudices is not going to be threading too thin) racist?

"jewish" has, at least historically, had both an ethnic and religious component; which is why anti-jewish could potentially be described as racist, depending on the time and circumstance.

this is not really the case for islam or christianity; use of "racism" there is a typical "postmodernist race theorist" fallacy used to shut down discussion and avoid addressing arguments.