It's not quests and loot that decide if a game is an MMO, it's the the different types of people playing it.
What do World of Warcraft and Minecraft have in common? Well, not very much at all, except for the fact that they're both MMOs. In Issue 288 of The Escapist, Joseph Bishop says that MMO players can be sorted into four broad categories, and not only does Minecraft support all four, they're uniquely adapted to suit their sandbox environment.
The four archetypes, as presented by Bartle, are as follows: Achievers, Explorers, Socializers and Killers. He goes on to compare them to the four suits in a pack of cards, with Achievers being Diamonds, Explorers being Spades, Socializers being Hearts and Killers being Clubs. Diamonds seek treasure in ever-increasing amounts, Spades constantly attempt to unearth knowledge, Hearts tend to empathize with fellow players and Clubs, well, hit people.
All four suits are represented within every server population with perhaps only Clubs being slightly less-so ... Even then, Hearts could be argued to be the most represented of the suits and are not entirely exclusive to their single categories. In fact, the test that was derived from Bartle's work actually allows for a total of two-hundred percentage points in order to allow for varying degrees of interests along different lines of player archetypes.
For example, it's rare to have an entirely pure player of any one suit. A pure Diamond wouldn't particularly care about Hearts recognizing their achievement; it would be good enough to know that they did better, built bigger, taller or more interesting things than anyone else. Typically, Diamonds have a bit of Heart added to them that showcases this affinity to brag. They don't just want to be the best. They want everyone to know they are the best.
It's not just the Diamonds - or the Achievers - who are busy in Minecraft. The Spades are, appropriately enough, busy digging things up, Hearts are sharing their secrets online and Clubs are indulging in architectural griefing. You can read more about it in Bishop's article, Achiever, Explorer , Socializer, Killer."