There are many, many reasons a fandom becomes big or assumes the form it ultimately does, including a combination of luck and visibility...although that's not quite what I asked. What is it about the properties themselves? What do they all have in common?
According to the unofficial history of Star Trek by Bob Justman, the first Star Trek convention was far from a spontaneous gathering. It was run by Gene Roddenberry himself and he rustled people in from all over. In other words, part of the reason Trek fans are so big and organized is (partially) because of talented huckster Gene Roddenberry's talent for self-promotion.
Lord of the Rings was a low seller in a niche genre that didn't even have a name yet for an entire decade, released at the same time as now-forgotten Thomas Burnett Swann's Minotaur books or Mervyn Peake... until LotR was rediscovered by the 1960s counterculture. The hippies found the story's themes of hostility to modernism, love of nature, and longing for an imaginary past were all right up their style.
And on a lighter note, part of the reason furry fandom is so oddly sex crazed and gay friendly, is because the first furry conventions in the early 1980s were put together by a flamboyantly gay swinger/S&M lover who actually invited just released prisoners from jail to show up! This jailhouse gay and S&M attracted others into the same scene, repulsed others that didn't care for it, and in general set the tone for years later.
Okay, okay, back to my original question: why do some fictional worlds inspire so much devotion and others don't? What do they all have in common that get people to still think about them? After all, the hippies came and went, but Lord of the Rings is still around, and there are still kids in high schools today writing notes in Dwarf runes.
If you look at the really BIG fan groups out there, like Star Wars, Star Trek, Lord of the Rings, and the Marvel and DC Universes, you find that people are so obsessive about them because these worlds have built up enough detail that it's possible they're engrossing and "full" enough to capture the attention.
The more real a story feels, the more engaged you are. Just like significant descriptive details in a book bring a scene to life, the richer and more developed a world is, the more minutiae fans can collect like stamps, the more devoted your fans will be.
Here's a secret: hardcore fans like their world and devote so much time to it because they think it's real.
…Okay, that's obviously not literally true, except for a tiny handful of seriously mentally ill people.
But really involving worlds and stories are set in places that make you FEEL like they're real, have verisimilitude, that the stories don't end the instant the book is closed or the movie ends. There's plenty going on to make you think about it later on, or talk about it with friends aneven have fights over nitpick details, and curiosity over some of its unexplained mysteries. This is why fictional worlds can be such an engrossing diversion. This is why people get very curious and so very passionate.
We fans do this not because we think the story is literally real, but because we like suspension of disbelief. Because when people tell us a story, we want to believe it. That's the difference between a liar and a storyteller: people are by default suspicious of liars, but want to believe a story.
Even disagreements and fights between fans are based on the idea characters are independent of us readers/viewers, solid and concrete. People have disagreements about whether Spider-Man would or wouldn't behave a certain way. This disagreement is only possible at all because everyone agrees there is a "correct" way for Spider-Man to act.
Okay, now, here, we've reached the first of definition of continuity:
Continuity (or canon) is just another word for verisimilitude, a consistency that makes the world feel more real, by making the world feel as if it continues beyond the boundary of the specific narrow focus of any one single story.
An individual story, by definition, have a laser pinprick focus and economy. Stories can't be "about" everything. Look at a map of Middle Earth and see how much of the place our heroes actually DIDN'T go, even in a story as sprawling as Lord of the Rings.
The purpose of continuity and consistent canon is to dispel the artifice of stories, that a world is not just a series of empty plywood "Potemkin villages" that live when tourists pass by and are yanked down just behind us when we leave.
"It's just a story"
Contrary to popular belief, being a fan, even a detail oriented one, is a good thing for the same reason any hobby not pursued to the exclusion of responsibilities is a good thing: because you get out what you put in. Cool detachment eliminates the entire point of a whole story, where you want to be passionate and care about what's going on.
I don't get that argument. If you don't care, why read a story at all? Why read a series long term?
This is also why fans are fascinated by cross-overs. The fact that Spider-Man can swing on the rooftops of Marvel Manhattan and cross paths with Daredevil going the other way makes the Marvel Universe feel more real.
Why Do Fans Like the Obscure?
This is also why fans get tickled pink when obscure characters and minutiae show up, why Star Trek fans smile a bit when a Next Generation episode mentions the Tholians in passing, or when people are delighted to see a minor character like Stingray or D-Man in the background of an Avengers mission.
Those of us that aren't children know, but only at an intellectual level, that stories aren't real, and are artifices with a very narrow focus, that the Tholians were just created to be bad guys for one episode way back when in the old show, and Stingray was an Avenger only used in one story that needed an underwater guy, and that a good portion of super-villains, especially unpopular ones, will never be seen much again after their first story.
Mentioning obscuros again dispels that artifice we're intellectually aware is there ("this is just a story"), and reminds us, to our joy, that the world is real, that these characters are still around and doing something even when we're not looking. The Tholians are out there, somewhere, up to mischief. Stingray can be called up at any time by the Avengers when needed.
In short, the world goes from something we like to something we believe.
And that's the difference between just liking something, and being a fan.