The way i look at it DF and Minecraft are incredibly similar - in the way that most games are.
Lets break down what Minecraft - And Dwarf Fortress - is.
I play minecraft in a specific way; I make a world, and then i build a story in that world. Im a shipwreck survivor. Im a time traveler gone wrong. Im a location scout for a future Castle-Building team that never arrives. Im a exiled dwarf digging out an existince in a hostile world full of monsters. And, most recently, in a multiplayer foray, im a Neighbouring Priest in a suit (In the Vein of Flanders from the simpsons) who collects taxes (By Force) from the other citizens of the world: A 27-times divorced Orphanage owner, Her Husband (For the 27th time!) and their Adopted Man-Child who previously burnt down the Church, The Forest, a Swamp, His Parents house and Himself.
That Exiled dwarf finds a village and finds a community to trade and live with, finding reasons to keep living. That Location-Scout takes it upon himself to build that castle in order to make a safe haven for the caravans that the kingdom relies on. That Priest gets the Wife, Husband and Manchild to build that new cathedral (Out of Stone) further making the world greater, only to kill himself in boredom after the church is complete.
I go into the world; Generated progressively and randomly. Thats all it does. But I... I make the story.
Im not the only one to do such things; Look at all the adventure maps and Youtube Channels - Look at the Yogscast. People who make livings from just being in a world they create. They created a world in which a powerful monster seeks to destroy, while a Spaceman and a Dwarf fight futilely against him. They make friends, and enemies, they discover lost secrets and ancient tombs. They lie, they joke, they cry, they get scared and they even die. But the Thing is - They do it. Not Notch. He Didnt put that story there.
But i ask you, Bay12, is that different from Dwarf Fortress? From Toady?
What about Boarmurdered? Did we not create a world around that one simple fortress? A Hole in the ground that happened to have elepants nearby? We have Saknis, we have Caacme, Tholtig Cryptbrain and so much more. We have little stories. We all do when playing Dwarf Fortress.
We have those forts that 7 dwarves set out to make in order to colonise that island. That fort that is a frontier defence. That Fort that may be the last stand for a falling civilization. The City that will outshine everything else for hundreds of years. The Death Pits. The Hell Holes. The Glass Towers. The Pyramids. The Lost Temples. Fallout-Esque Vaults. Kobold Caves. Volcano Doom Lairs.
I embark with my seven lost merchants; Unskilled dwarves lost in a stinking bog, out of food and drink. The only solution is to Dig down and hope to be rescued. But when they strike rich veins of silver and gold, they end up wanting to stay, becoming proud of their new home, and becoming rich through hard, honest Dwarven work. I have the Seven Dwarves that are the final refugees from a fallen civilization, lost and scared, with only a single unarmed militiamen for protection from the harsh wilds and invading humans. Do they hold? Do they rebuild? What becomes of them? Where do they go from there? What happens when the Militiaman is crippled by a Lion while out hunting for the only food they can get their hands on? Will The last two females be able to repopulate the world with Dwarves? Will the Farmers finally grow a successful crop of Plump Helmets before the last of them starve?
How about the time when the Vampire arrived in the fortress, quickly killing two dwarves without leaving any trace? Can the militia find this dwarf in time, before another murder is committed... Only for it to be revealed that it was the Fortress Guard Captain who was the vampire all along? What about that time when a Goblin ambush arrived, leaving three dwarves trapped outside the fortress with no hope of survival? Do you remember how one of those dwarves - only just a young dwarf of 17 - enraged and charged the goblin invaders, leaving enough time for the other two to escape to safety, only for one of the survivors going berserk at the thought of the militia not going to save the heroic young dwarf?
We don't make Fortresses in Dwarf Fortresses. We Make Stories.
I think this is where Notch found his inspiration.