There's not really a "genre label" for this kind of gaming, simply because non-interactive games aren't much of a game genre. Most of the ones that exist are either silly idling games (Idle RPG/Progress Quest/Idle 2), bots (like this one that plays a roguelike by itself), or more AI experiment than game (Conway's Game of Life, for instance).
I think that "Zero Player Games" or "bot game" is about the best you can do. I did find some interesting articles when I searched for "games that play themselves."
There's also a subgenre of the strategy genre where you'll find games where you can only interact indirectly with the game world, with games like Populous or Evil Genius. This subgenre is generally called "Artificial Life" or "God Game".
Some games are capable of playing themselves, in a demo or "attract" mode. Arcade games are especially fond of this, as it attracts people to the cabinet and shows them what they might get for dropping in a quarter.
If we consider things like ProgressQuest games, then screensavers might be considered non-interactive games as well (or in reverse, non-interactive games might be screensavers). There was one screensaver for Win95 that played a little maze, for instance. Of course, there's also the iconic After Dark package that not only featured a number of things you might consider to be non-interactive games, but also some screensavers that were fully playable, like Lunatic Fringe.
Part of the problem is that we already have many non-interactive forms of digital entertainment. We give them many terms, but one is "movie." You could consider a movie like Scott Pilgrim vs. The World to be a "game that plays itself." The protagonist even levels up, has extra lives, and fights a final boss. Or, take the movie Clue, which went so far as to include multiple endings, and each time you went to see it there was a chance you'd see a different one. (I realize that's tangential, but it's sort of an interesting aside to your question.)
"Non Games" is a similar term, but it's more for games that have no goal, like Mario Paint. These games ride the line between game and application.
That's called "shared screen", as the players share the screen.
You don't see this term all that often, as not only are these games not particularly common, but it's also usually not necessary to make the distinction. People will say "split-screen" if the players split the screen, and not mention it otherwise.
Here are a few examples of this term in use:
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The term for this is "pixel hunt" although I'd stop short of calling it a genre. It's more a (often considered negative) game design element.
Hidden object and adventure games often employ this element, as do "escape" games.