Technically it has many elements of horror from real life horror (being eaten alive by a shark is definately something everyone considers horrible fate) to horror tropes (violent death, use of blood/gore, jump scares) however as the film progresses (Quint's death aside which is a horror film death if ever their was one) it becomes more of a tense hunt thriller and overall it feels more like a thriller but there's enough horror content to make a case for Jaws being in that genre - same with The Sixth Sense.Jaws is considered horror? I always just considered it a thriller
TBH these days I'm mostly anti the idea of genre - it's just a way of trying to classify material that often combines themes and styles that are seen as belonging to a genre.
I guess libraries and mass market like the idea still but personally I think if you like films point blank (or novels) the idea of genres becomes merely a filing system and an imperfect one at that.