So, because I think this is really the ultimate form of game, I tried to make a list of games with a somehow similar feel to KODP. What that means is, that those games have RPG elements (i.e. you control a specific group or a single character and are heavily immersed into the game's world) and focus on emergent storytelling achieved by dynamically generating overarching plot developments.
I came up with 13 such games, that I have played. But also, none of them really nail the gameplay as perfect as KODP and I'll try to sum up why that is. This list will contain games already mentioned here, but I'll include them just for the sake of completeness.
7 Grand Steps: Currently still in beta but already a great game. Feels like a solitaire board game pretty much focused on mechanics, but the emergent stories are definitely there (you guide many generations of a family through copper, bronze and iron age), also in the form of textual events sometimes with multiple choices. In the end the game is quite simple and can get a bit samey after a few playthroughs (there is much variety in the game, but not necessarily in the single ages/social levels).
Academagia: Kind of a Harry-Potter-simulator (you're a student at a magic school). It tracks a crazy amount of stats and skills, but in the end everything feels quite bloated and unclear. Also, the events mostly don't really manage to create tension. Most of the time is usually slowly building up the skills anyways (quite similar to those Japanese life sims). Maybe future titles in the series ("Year 2" is in development) will solve some of the problems.
Dwarf Fortress: Probably the most ambitious game there is. It randomly generates a giant world with a detailed history, mythology etc. Then you as a player lead a group of dwarves to build a fortress in any region of this world. Also, there's a adventure mode, where you play a single dwarf and travel through the world. Very open sandbox gameplay with near endless possibilities. Problem: This game will probably be in development forever and up to now has no graphics (everything is ASCII) and the interface is exhausting at best. Still, quite an impressive piece of software.
Crusader Kings II: This is the most rpg-like of the Paradox grand-strategy-games. You control a specific medieval character and lead his dynasty through many generations. There are random events and certainly emergent stories (family feuds, year-long conflicts, alliances through marriages etc.), but after all the game feels pretty conflict-focused and at times pretty unclear about what's going on. It's definitely recommended though if you're interested in KODP-style games and are just a bit into traditional strategy games, too.
FTL (Faster Than Light): The recently very successful spaceship-roguelike has random (textual) events, characters building up their skills and much variety in possible routes your journey can take. Apart from that, it's obviously not as epic as other games discussed here as it focuses on short playthroughs (30-60 mins). Also, the random factor is (as for many roguelikes) extremely strong and can be quite unfair at times. The game is also pretty combat-centric and I don't think the combat system is that brilliant. It's still quite a good game though.
Lords Of Midnight: This is a really old game (much older than KODP) that was recently remade for iOS. While the graphics didn't change that much, the interface works quite well in today's standards. In the game you control a set of characters in a somewhat medieval fantasy conflict against the dark side. You can either win by stealing the ice crown (more of an adventure/stealth game) or by war (more like an early grand strategy game). This game can certainly generate stories (although the world itself is completely fixed) but it feels pretty dated and might at best be somewhat of a spiritual predecessor to something as timeless as KODP. The controls if you're going for the victory by war can also get quite tedious. To me this game is more interesting from a gaming historical point of view than of a today's player's perspective.
Mount & Blade: This is pretty much Pirates in a medieval setting. You make your way through an open world with tons of things to do. While the idea is certainly great, it feels pretty rough around the edges at times and could really use some more polish. Also, the combat in the game is action-based (you directly control a single character from a 3rd or 1st person perspective), which might not appeal to everybody interested in the kind of games we're talking about here. It's still an interesting effort and I'm eager to see what the upcoming sequel will bring.
Pirates!: Who doesn't know this game? Roam the caribbean for treasure, glory and fame. A classic that was remade a few years ago and is even available on iOS today. Problems one could have with it: It's quite minigame-centric and has some dexterity elements here and there. Apart from that it's really a game any gamer should have at least tried at some point.
Organ Trail: A zombie survival game based on Oregon Trail. A remake of this is currently pretty successful on iOS. In the game you're controlling a group of survivors traveling by car along the road. There are (simple) random events and quite a few decisions to make. The problem I have with it are the somewhat clunky dexterity-based minigames (executed while scavenging, fighting etc.), that can make up quite a bit of game time.
Star Traders RPG: An open-world space trading simulations available for Anroid/iOS. Somewhat similar to Pirates as you can also choose to fight and steal your way to fame in the galaxy. It features almost no graphics and text-heavy interfaces. Also, it's pretty hard to get into the game and understand all the interconnected systems. If you invest yourself into it, it's pretty interesting thouhg, even if the "standard" trading gameplay can get a bit tedious.
The Lost Tribe: This game is often mentioned in connection with KODP. While you also lead a tribe here, it's really not nearly as complex or sophisticated and also more of a journey-game. Considering this game is a few years older though, it's pretty interesting and was probably ahead of its time. It obviously didn't age as well as KODP though.
Weird Worlds (Return To Infinite Space): Somewhat a spiritual predecessor to FTL. You're traveling through space from planet to planet and everytime some random event happens (you meet an alien race, discover artifacts etc.). The number of things to discover isn't that large though and you might have seen everything after a dozen playthroughs (each might take around half an hour). Also, the combat needs some getting used to and can be pretty random and even frustrating at times, but then again the playthroughs don't take too much time really. It's quite a good game and definitely worth a shot.
Zafehouse Diaries: You're in control of a group of survivors during the (once again) zombie apocalypse. The interesting part: The game tries to simulate group dynamics as relations of every character to one another. Gameplay-wise you give orders according to the characters' strengths and watch everything play out in a purely textual diary (reminiscent of KODP's sagas). Again this is quite an interesting idea, but the execution is somewhat lacking: the events feel samey after a few read pages while the outcomes are sometimes frustratingly random. This game could really use some more content and polish and become something much greater.
So, the bottom line: There's just nothing like KODP in the current gaming world, but some interesting forays into the area of emergent storytelling have definitely been made.
Let's have a look at some promising future releases besides the already mentioned sequel to Mount & Blade.
Banner Saga: This is described as KODP meets Oregon Trail. That should sound interesting enough already. It will be more focused on tactical combat though, but still might turn out pretty great in terms of storytelling. We'll have to wait and see how much variety the random events really hold.
Project Zomboid: Yet another zombie apocalypse survival game? Yeah, but this could really be the ultimate experience of the setting. While the game is certainly more action-centric and controls kind of like a Diablo style RPG (there's a tech demo available and early alpha access through pre-order currently), there are some incredible features in the works at the moment. Most important of all is the so called "meta game". What that means is, the game will generate stories with an open event-driven system. Those events play out in the whole game world as "meta events" while your character walks around somehwere else. If you're there to encounter the events though, you can witness, take part or completely change the world's story. I followed the game's developemnt quite closely for some time now and am confident the devs will be able to pull this off and really create a completely unique experience.
Unwritten Passage (That Which Happened): The concept behind this game reads incredibly interesting. The first lines of the Kickstarter page say: "Permanent decisions and infinite variety. Create a nomadic tribe and guide them across a randomly generated tundra to meet its God." Wow! There's no real gameplay yet, though. But it's definitely promising and emergent storytelling seems one of the main factors considered in the game's design.