Bioshock is absolutely a horror game. The residents of Rapture are meant to be scary... the noises and feeling of dread, the environment / sounds invoke are strong horror vibes.

It's just not as scary as it would have been b/c of the plethora of abilities afforded your character to deal with the denizens of Rapture.

Restrict the player to what they can carry at any given time and give them less resources and you'd have a Survival Horror game.

But as it stands I'd say it lands closer to Action Horror.