It's an interesting question. I think that the rationalization
behind modern antisemitism has been racial (remember that the Nazis were obsessed with social darwinism). But its long history in Europe has been both religious and (perhaps more importantly) cultural. The first "genetic" antisemitism began in the Spanish Inquisition in the 1490s. In this case, even the conversos
(Jews who had converted) were persecuted.
The Nazis used many of the same stereotypes against Jews that people had been using for centuries, but their rationalization was racial (even though there isn't any such thing as a Jewish "race" to begin with).
It's ironic, because many of the stereotypes about Jews were foisted upon them extrinsically. Why were so many Jews in banking and finance? Because the church made usury a sin, so Christians couldn't do it for fear of excommunication -- but Jews filled that niche and Christians were more than happy to do business with them. Why were so many Jews in medicine? Because all of the universities in medieval and early modern Europe were Christian and medicine was the one secular field they could study.
Muslim antisemitism in the modern world is wholly political. It's neither religious nor racial / ethnic. Muslims had a long long long history of benevolence towards Jews really until Zionism came about.