Even later on the wiki page , the influence of survival horror becomes a lot more apparent

The game's lead level designer was Bill Gardner.[37] He cited Capcom's survival horror series Resident Evil as a significant influence on BioShock, stating there are "all these nods and all these little elements that I think you can see where Resident Evil inspired us". The team were particularly influenced by Resident Evil 4,
including its approach to the environments, combat, and tools, its game
design and tactical elements, its "gameplay fuelled storytelling" and
inventory system, and its opening village level in terms of how it
"handled the sandbox nature of the combat" and in terms of "the
environment

It's also the spiritual successor of system shock 2, which is classed as a action role playing survival horror game.

Bioshock has more action in it though, hence that's probably why they classed is as an fps game, but that's a very broad description, call of duty is also a first person shooter.