You can slice up the universe of interactive media along many different dimensions, and genre is only one of them. But in order to be able to talk about it sensibly, we have to define what these dimensions are.
Fiction book genres are based on subject matter and to some extent, emotional tone. There are techno-thrillers, westerns, romance novels, mysteries, spy novels, historical fiction, and so on. The book's setting plays a role in defining its genre.
Film is much the same, except that far more films than books are comedies, and just about any other genre of film can be made into a comedy: Blazing Saddles and Rustler's Rhapsody were western comedies, for example.
Video games work differently, as we know, and so far as familiar cases are concerned, it's easy to see. Video game genres are determined by gameplay: what challenges face the player and what actions he takes to overcome those challenges. So we clearly have sports games, shooter games, racing games, and so on.
Genres actually exist in a hierarchy. The Entertainment Software Association's 2009 Essential Facts About the Computer and Video Game Industry explicitly identifies (using data from the NPD consumer research firm) a number of what they call super-genres.
For example, sports games can obviously be further subdivided by sport, and racing games by type of vehicle. Tactical shooters emphasize real weapons and quantities of ammunition, while games like Quake, which is mostly about speed, do not.
Genres themselves are fluid. More and more games are hybrids of the traditional genres: modern role-playing games often incorporate physical coordination challenges more typical of fighting games, and action-adventures are now more popular than either the pure platformers or the pure adventure games they borrow from.
And in the current explosion of creativity, it may be difficult to assign a game to a familiar genre, as with Passage. But that's all right. It's still the case that gameplay (which determines genre) is a separate dimension from the other dimensions that define a game. I'll look at them next.
A game's setting is independent of its genre, which I think confuses people who are more familiar with books and movies. Westerns and science fiction books are in different parts of the bookstore, but that's not true of games.
A shooter is a shooter, whether it's set in the Old West or on Mars or anyplace else. A player who enjoys shooters will probably enjoy one no matter where it's set, if it's well-made.
Likewise, you can have completely different genres in the same setting. Both Gun (a shooter with RPG elements) and Oregon Trail II (a management game with a hunting mini-game inside) are set in variants of the Old West.