The mass media has a tendency to treat all games, or all games of a “genre,” as the same. Consider though, that while Bioshock and Counter-Strike are both First Person Shooters – they are entirely different games. Bioshock is decidedly a single player game, that has little penalty for a player's death, no real way of “keeping score,” and a heavy focus on the game's plot. Counter-Strike is an online multiplayer game where two teams compete under clear win and loss conditions.
Games that focus on a story-like experience or journey are similar to books, movies, and group roleplaying activities. Games that focus more on competition and quantifiable scores have more in common, structurally, with sports and traditional games. You just can't be a world-champion Bioshock player, much like you can't play a game of solitaire for the story. (Many games have multi-player, challenge, or campaign modes tacked on for additional value – but these modes are more like additional games within a package.)
First Person Shooter (FPS for short) is considered a video game genre, but the term only tells you about the controls and the camera angle. Outside of these genres, games have the potential to be more diverse: The only thing a Point and Click Adventure game may have in common with the latest copy of Madden is that both games can be played on some form of computer.
The bottom line is: there are so many “genres” within the video games umbrella, and so much potential for differentiation between games in a given genre, that we can not afford to lump all games in the same category.
The only thing these “games” are guaranteed to have in common is that they all run on electricity.