IMO, Cyberpunk is a style of visuals and setting rather than a genre. The colourful, rebellious, self indentified identity of punk with futuristic, sci-fi elements. It's like not like video games are the only expressions of cyberpunk. For example there are several movies, like Tron and The Running Man that I'd describe as cyerberpunk as well.
In terms of games, and as you've shown, first person games seem to be a popular choice with this aesthetic in recent years for reasons I can't quite figure out. But still, I have seen RPG's, FPS's, side scrollers and even racing games with that aesthetic theme. There are limits, it's not like all genres suit the aesthetic. Horror is one that strikes me as one that really wouldn't work.
But that's not the point. Cyberpunk is an art style. An art style that suits some types of games and stories more others, but an art style none the less.