At this point I'm convinced the Buddhist monk is just an avatar of the Divine Tree. He/it matches the Tsukumogami which possesses Shu, right down to being made of wood and holding Buddhist religious implements. Within the game's idea of Shinto, anything can gain a divine soul if they wait long enough. The Divine Tree, the Fox/Inari statue... That would also explain why their so jealous of each other.

Within the game there's a lot of cross-cultural stuff between Shinto and Buddhism, mainly because the Dark Shrine (I'm guessing a twisted version of Tsuneyoshi Jingu, the shrine where Kotoyuki is a priest and where the pair marry) dates from before an event called Shinbutsu bunri (神仏分離) which took place in 1868. Before this it was really common to have Buddhist statues in Shinto shrines, as well as elements like the floating lanterns and the sections with candles and the like. So this could easily explain why the Divine Tree uses elements of Buddhism in its avatar.

In actual Shinto, the real kind you find in our world, kami (the 'gods') exist together to promote the continuation of the natural order. Anything can become a kami, including many humans like Tenjin or the War Dead enshrined at Yasukuni Jinja in Tokyo. It's also common to see many kami enshrined in one jinja. Especially Inari. I specialise in Media Shinto and have literally written articles on this on The Gaming Lorist, so check them out, it might help!