There is a line in conjure animals that "Each beast is also considered fey, and it disappears when it drops to 0 hit points or when the spell ends", so things like the banishing effect of divine word, protection from evil and good affect them. I would also consider the conjured animals to be immune to dominant beast, although being a kind DM, I would let any spell that affects beasts that the caster wanted to affect the beasts (like beast sense or beast bond) affect the conjured critter (also being the key word in the description).
Of course, there isn't a "spirit" type in 5e, so RAW there is nothing to compete with the "any humanoid" part of the description of berserker. I would go with RAI (and RAF), and since the DMG says there are elves and dwarves that are celestials (or to borrow the idea from conjure animal, they also have the celestial type in addition to the humanoid type) running around the Upper Planes, the berserkers are probably celestial versions of humanoids as well (if the horn holder's player had an inspiration point, he/she could request the berserkers be celestial goliath berserkers, celestial half-orc berserkers, or celestial any other humanoid race berserkers).