This discussion contains spoilers for Hardcore Henry.
It may sound like quibbling, but describing the film Hardcore Henry as being like a First person shooter seems less accurate to me then describing it as a first person on rails shooter. The first person shooter is obviously the major influence on the style of the film. However, because of the nature of cinema, it ends up resembling an experience more akin to playing an on rails shooter. This distinction seems useful to me because I think that the latter observation about the film speaks to its thematic concerns more clearly than the former observation.
It is true that shooting a film from a first person perspective creates a kind of intimacy (and, perhaps sense of complicity) with the world and characters of a film that is similar to the intimacy and sense of complicity that seems similar to playing an FPS.