How does "denomination" interplay with schools of thought

From what I've read it seems like the distinction between denominations is small due to the concept of Ishtadevata. I heard someone said that Krishna was devoted to Shiva and vice versa so that aspect wasn't a big point of contention. But how does it interact with philosophical schools? While it seems like Vaishnavism and Shavaism aren't really competitors, I get a different impression from the schools. Is there a larger divide between a vishishtadvaita and a dvaita Vaishnavite than between people of different denominations?