BioShock 1? Without a doubt it's a horror game. It's basically a zombie game except they call them "splicers" and technically haven't died yet.

The lighting is spooky- the whole game is dark. Being underwater is another classic horror setting. There are unethical doctors who caused a lot of this, and one of the first areas you go through is the medical area, including surgery and dental areas. Classic horror tropes.

I don't just think BioShock is a horror game, but one of the most influential ones in history. There have been plenty of conversations on the internet about how you can technically find examples of other games doing the things that BioShock did earlier (System Shock, Half-Life, I could go on). But it was the first game that combined all of these elements and executed them well, and it led to a huge boom of what I consider to be heavily Bioshock-inspirwd horror games. Markiplier became one of the most successful and wealthiest content creators on the internet by tapping into the flood of these things. Bigger games like Amnesia: the Dark Descent or the flood of "walking simulator" indie horror games.

BioShock 2? I have only played like an hour or two, but I do know it was made by a different development team and have heard it focused much more on the gameplay and multiplayer than the story. Still possibly a horror game but less so.

BioShock Infinite? I don't think I would call it a horror game. I would be open to arguments for that though. Perhaps in a decade or two people may start to group these stories that involve "multiverse" travel, or even just time travel, as a sort of existential horror. I would say Infinite has a case for that, but you might need to add tentacle monsters to fully cross the threshold into horror there.