Those are all Abrahamic faiths that derive from a single common theology, that of Judaism. As such, they have a great deal in common. Both Christianity and Judaism, for example, teach the Old Testament. Possibly Islam as well, though I'm not sure about that - after all, Islam teaches that Muhammad was simply the last in a long line of Prophets that includes Jesus and the Old Testament Prophets. I think that your argument would not fare as well were you to use Christianity and Hinduism, or Judaism and Buddhism.
Yeah, Christians and Jews really hate the Dalai Lama.
But, in all seriousness, looking at the Mughals and other Indian states suggests that (or the Manichees or Nestorians earlier), in at least some cases, cross-religious spiritual authority of some kind was recognized.
Again, for a Medieval thinker, spiritual discussion is
not relativistic as it is for us. We believe, or most of us do, that the most direct method to objectively debatable reality is probably the scientific method. Medievals, mostly (a few interesting dissenters among Christian heretics and some Muslim philosophers come to mind as counter-examples) thought that spiritual authority was a direct and objective thing. It's like if two scientists with radically different approaches to, say, evolution, just said, "Well, your model can be right for you, mine is right for me; ultimately, who's to say?" No, those people have a common idea of a methodology for debate and argument.
Similar things existed for Medieval philosophy and theology; common understandings of debate (such as the need for classical precedent in much of the Christian and Muslim intellectual world). They wouldn't say, "Don't bring your religion in here!" they would say, "Your religion is false, for we all know that Plato has said XYZ, which you erroneously interpret..." Now, of course, we don't have Plato. But the general idea should remain: that our characters, being Medieval, will not regard religion as a purely subjective matter, or something without recourse to objective mediation.
And I have often wondered why Mathurin doesn't take the Dean of Theology position.