I'm not sure if you're fighting against a concept, or merely against your own ridiculed vision of it. Obviously, not every person in one civ was at war with another. But then, it never is so about any larger group of people.
Orthodox and catholics? It really wasn't that important... In Byzantium, perhaps, especially after 1204. But even when it comes to Byzantium, they were involved with the crusades. The crusades were called out of orthodox emperor's instigations, he was supposed to help them on the way to Jerusalem and would have, was it not for a certain person (was it Steven of Blois?) didn't convince him and crusaders were already defeated. Later on, during Manuel Komnenos' times, but even during Kalojoannes' reign, the empire considered itself a sort of patron of Outremer.
Shia and sunni? Hardly relevant. I don't see your point.
Of course, that doesn't mean that all christians were bent on killing christians and vice versa. There were muslims participating in christian holidays, there were crusaders befriending local muslims, there were traders on both sides etc. Obviously. But that doesn't change the fact that muslims, at least were aware of their obligation to spread the borders of islam, and christians were aware it is their duty to protect / conquer the Holy Sepulchre. Their personal or communal interest may have made them act against these moral obligations, but it was not frequently denied; Venetians trading with muslims were well aware they are doing wrong and felt obligated to visit sanctuaries and ask God for forgivance.