Sarah's real mother (as seen in gossip rags) was dating an actor, depicted in a news article by a photograph of David Bowie. Since the entire film is basically Sarah's fantasy, of course the Goblin King would resemble someone dating her mother! The Novelization is more blunt about this, as it goes further into Sarah's backstory via a Flashback or two.
Ambrosius is another name for Merlin.
The movie inverts all of the fairy tale books seen on Sarah's shelf at the start of the film.
The Wizard of Oz: Sarah sets off into the Labyrinth, which has no clear path (the opposite of the Yellow Brick Road, where you only have to stay on the path to be safe). She meets Hoggle (all brains), Ludo (all heart), and Sir Didymus (all courage). Dorothy pledges to help her companions on their own quests, where Sarah's companions aren't really seeking anything and choose to help her on hers. She must rescue her little brother Toby instead of her dog Toto. The Junk Lady corresponds loosely to the Wicked Witch of the West, who wants to keep her there and make her one of The Junk Lady's kind, instead of killing her and taking her shoes. Dorothy's power is her innocence, whereas Sarah's is gained through maturity and coming of age. The poppy scene corresponds loosely to the peach scene. The aesops are inverted: "Every now and again in my life, for no reason at all, I need you [i.e. imagined things]" versus "If I ever go looking for my heart's desire again, l won't look any further than my own backyard, because if it isn't there I never really lost it to begin with." Almost the first thing we learn in Oz is that good things are beautiful and bad are ugly, while we learn the opposite at the start of the Labyrinth with Hoggle and the pixies. To defeat the Wicked Witch of the West, Dorothy must first enter into subservience to her, whereas to defeat Jareth, Sarah must utterly reject subservience to him. The parallels go on, if you're looking for them, and many (like the sidekicks) seem less-than-coincidental.
"Cinderella": Sarah sees herself as a Cinderella figure at the start of the story. She spites her stepmother and half-brother and describes herself as "practically a slave," while her stepmother complains, "She treats me like a wicked stepmother in a fairy story no matter *what* I say..." But of course, the "wicked" step-mother and half-brother are decent people, while the sparkly prince is the villain. The ballroom scene, which is a huge shout-out to Cinderella (especially with that white ballgown), is sinister rather than triumphant. And instead of spending the story trying to escape an "evil" family by marrying a "perfect" prince, she's trying to rescue and reconnect with her family, while avoiding the lure of the seductive prince. Also, while Cinderella starts off as a slave, and ends as a princess, Sarah, in a sense, is a "princess" from the start, as emphasized by her Meaningful Name, and Jareth ultimately wishes to make her obey him completely.
"The Little Mermaid": The mermaid leaves her family to enter a race against time, to be with a prince she doesn't even know. Sarah on the other hand is in a race against time to rescue a family member, and resist the urge to leave her family for the prince. Also, Sarah's dog Merlin (and by extension, Ambrosious) resembles Prince Eric's dog Max, from the Disney version of "The Little Mermaid" (though this is a coincidence, since Disney's "Little Mermaid" came out in 1989, three years after "Labyrinth").
"Beauty and the Beast": Many people have noted that Jareth's ballroom attire resembles the Beast's from the famous Disney version of Beauty and the Beast. (This is also a coincidence though, since Disney's "Beauty and the Beast" came out in the '90s, long after "Labyrinth.) While Belle must learn not to judge a beast by his looks, Sarah is faced with the opposite lesson of not judging a handsome but sinister prince by his looks.
The major characters in the Labyrinth stand for family members Sarah must learn to deal with in a mature way.
Ludo, the slow-minded innocent beast, is like a baby, who Sarah must be patient and gentle with (like Toby).
Sir Didymus is a wild, nasty-tempered and single-minded little creature, like a young child - but ultimately caring and unwaveringly loyal, like one's own child or younger sibling.
Hoggle acts harsh towards Sarah, but secretly cares for her, like her father and stepmother. Sarah must learn to look past his harsh words and see that he really cares about her.
Jareth stands for adults who are unfair towards Sarah. The way to deal with them is to remind herself, "You have no power over me."
The two door knockers who are always arguing over trivial things? Since Sarah's parents divorced, she no doubt spent her childhood listening to them argue and insult one another. She must learn to tune out their arguments, and try to deal with each of them as patiently as possible.
The Wise Man and his phallic hat probably stem from her relationships at school. Her school life isn't seen or mentioned, but it doesn't have to be; we can safely guess that she has trouble accepting attention from boys, with her fear of growing up, and the fact that (as her stepmother says) she's never had a date. The hat (which is clearly shaped a male symbol) wolf-whistles at Sarah—not in a taunting or cruel way, but in a light humorous way—and Sarah gives a small smile, unsure how to react to this attention. The hat stands for the boys at her school, who want to get close to her, and who she's shy around. The wise man stands for her teachers, who probably want to help her mature (but are always interrupted by heckling class clowns).
The dancers in the ball scene are not just an illusion, they are Jareth's goblins in disguise. If you look carefully, many if not all of their masks match up with individual goblins seen following Jareth around. (For one example, the woman whose mask has a bird beak and a unicorn horn matches with a goblin who has a bird beak and a horned helmet.) If one follows the theory that every goblin was once a human child who was wished away, then their human appearance in the ball scene might reflect how they "really" look, without the goblin curse on them.
The word oubliette comes from the French word for "forget". It really is literally a place you put people to forget about them.
Sarah and Hoggle are actually very similar people and go through the same personal journey. Sarah begins the story acting very childish, selfish and mean - ugly on the inside. Hoggle begins the same way, although he is also ugly on the outside. Together they learn to be kind and selfless over the course of the movie.
Everything that happens in the play Sarah is rehearsing happens in Jareth's world, including the kidnapping of a child. Everything he's done is what she wanted, which gives strong evidence that yes, it's All Just a Dream.
In Labyrinth when Sarah has her memories wiped, she is taken care of by a garbage hag, who wears a ton of garbage on her back. When she insists that everything that Sarah truly loves or needs are her toys and mementos, she piles them on Sarah's back, much like her own mountain of junk. This implies that she is trying to turn Sarah into another garbage hag, which in turn implies that the garbage hag might be another girl who failed to rescue her brother from Jareth. To make matters worse, you see that there are at least half a dozen others in the background, maybe more. Indeed, there may be hundreds or thousands, since when Sarah first ran into the garbage hag, she mistook her for another heap of junk because she was lying there doing nothing.
There's also the strong possibility that every single goblin in the Labyrinth (with the possible exception of the goblin king, himself) was once a human child that had been "wished" away.
Well, considering that the sequel comic (which is canon) explains that Jareth was planning to make Toby his successor....
What about the Bog of Eternal Stench? Supposedly, if you come in contact with its mud or water, you smell bad forever. While not addressed any further than that within the context of the movie, this presumably means a lifetime of rejection and loneliness until you're driven to madness or suicide. Or you become sort of a "bog monster" (not necessarily physically deformed, but people might call you that) who tries to trick or force people into the bog in an attempt to create companions (although Sir Didymus seems pretty friendly, and he claims not to smell the bog).
In the official tie-in sequel manga, that's answered. Remember how Jareth said that if Sarah kissed Hoggle, Jareth would turn him into a Prince? Well, he did. Prince of the Land of Stench! Apparently it's not exactly a blast.
Actually, Jareth tells Hoggle he'd turn him into a Prince... of the Land of Stench! when Hoggle responds with an incredulous 'Really?'