Is occult fiction classified as horror?

It can be, but it also possible for a story with occult elements to have no elements or horror in it, or any intent to inspire fear. I once came across a cutesie tween book series about girls who solve crimes and figure out if boys like them by using fortune telling. There are also romance stories that involve vampires, witches, werewolves, and even comedies that have occult elements.

So no, I would say that just the presence of the occult wouldn't make a story a horror story. The author has to have the intent to inspire fear in the reader.

Fiction could have an occult premise and be almost any genre.

I've even read occult westerns. Genre depends on how you write the book. Frankenstein is generally considered horror. So how about Young Frankenstein? Same monster, same life from death, but horror it ain't.