Because the characters were shown and later confirmed to come to life from dolls/trophies, with the "stages" being tabletops that transform into game settings, a popular and widely-accepted interpretation of the Smash Bros. universe is that it is created within the imagination of a person (most commonly a child, but possibly someone else like a wizard, Masahiro Sakurai, an ambiguous person, or even the player) playing with the toys and imagining fights between them, with Master Hand as that person's avatar in the Smash universe. A million different interpretations of this exist, from direct comparisons to Toy Story and The LEGO Movie to theories about the metaphysical aspects of the characters' existence and what it means for Canon Welding, and are a great way to break the base.
Going in the opposite direction, since the playable characters are merely imagined versions based on dolls/trophies in-universe, some fans claim that the Smash universe is similar to Toy Story where the characters are literally dolls or trophies that spontaneously come to life. This is an exaggeration of the Living Toys angle that was used to justify the nature of the series, and the bits of Canon Welding in 3DS/Wii U seem to disprove this interpretation. The Dark Worlds in Deltarune provide a better analogy than Toy Story, where various toys, games, computers and other inanimate objects appear as living beings in the Dark World and putting them all in the same real-world place causes them to all coexist in the same location.
Expanding on the above, Master Hand and Crazy Hand are near-universally believed to be how the trophies' owner appears to the Smash characters in their universe — drawing even more similarities to The LEGO Movie, where "The Man Upstairs" was seen by the LEGO characters as a giant hand and depicted as a god-like figure. Some people take this a step further and claim that the Hands are supposed to represent you, the player, and that the Smash fighters are brought to life by the power of your own imagination. On the other hand, some see them not as the creator, but the creator's gloves, putting Master Hand and Crazy Hand on roughly the same level as the other toys and figurines.
Master Hand and the Announcer are one in the same. This is because the announcer in every game shares their VA with Master Hand (and, starting in Melee, Crazy Hand), leading many to believe the casting choice is deliberate as opposed to convenient. Some fans will even argue the theory works best (or only works at all) if one of subscribes to the idea that the Hands are a child playing with their toys, thereby explaining the Announcer Chatter and why the first three games have a near total absence of it when it's time for the final battle.
Even predating Brawl giving all the fighters a shared world to live in through its Adventure Mode, The Subspace Emissary, is the idea that the fighters live in a giant mansion where they spend their downtime between fights, with the concept arguably popularized by Awkward Zombie. Early on, the concept was framed as a Reality Show/Big Brother-esque setting, but most post-Brawl works have since scrapped this in favor of the idea that Master Hand (or even Masahiro Sakurai himself) invites characters from various universes every few years to fight in a tournament, with the mansion being where they all stay for the duration of the event. While the "mansion" has no basis in the games, Snake's conversation with Colonel from his E3 reveal trailer for Brawl, followed by the regular use of characters actually receiving invitation letters in their reveal trailers starting with 3DS/Wii U would go on to lend strengthen this particular piece of fanon.
A few people take it a step further and give the Assist Trophy characters a place to stay as well, usually an apartment complex or a hotel.
Sometimes instead of Master Hand, comics may depict the house as being run by Sakurai.
Villain characters being Mean Character, Nice Actor, often in accordance with the above theories. There may sometimes one or two aversions depending on the fan.
To explain the Decomposite Character element of Zelda and Sheik from the fourth title onwards, fanon is that Sheik in 3DS/Wii U is actually the Ocarina of Time Zelda, since she's the only Zelda to canonically take on the disguise. Ultimate's version of Zelda and Sheik have something similar, with the latter frequently portrayed as the Twilight Princess version of Zelda, as she's the one who appeared in the previous two games, or even the Zelda from Breath of the Wild, based on Sheik wearing said game's Sheikah Stealth Suit.
Mario and Dr. Mario tend to be presented as coming from different universes altogether, which is not as far-fetched as it sounds, thanks to the franchise itself playing with the idea of alternate realities in Mario & Luigi: Paper Jam.
In stark contrast to the Sheik and Dr. Mario situations, Samus and Zero Suit Samus are rarely portrayed separately by fan works, instead simply having Samus alternate between wearing her Chozo Armor or not.
Captain Falcon is sometimes depicted with a variant of Pokémon Speak in fan works for Rule of Funny purposes, where he is only able to say his various catchphrases or extremely simple sentences that start with "Falcon". Not only is he fully articulate in F-Zero GX and the anime, but even Smash 64 (the source of his memetic quotes) gave him an additional written line in Engrish at one point ("Thank you for playing. Yes. I am Number One!"). Other fan works either have him speak proper English (like in F-Zero) or logically expand on his Smash speech patterns to have him speak Engrish at length. Others view him as being a Johnny Bravo-esque Elvis Impersonator.
A common idea is that Pikachu is a Legacy Character, with the one we see from Brawl on being none other than Melee's Pichu. This is due to Pikachu's Final Smash being Volt Tackle, a move that it can't learn but Pichu can, and Pikachu having Pichu's goggles in one of its alternate colors. What happened to the original Pikachu, and what's his connection to the current one, varies between fans. When Pichu returned alongside Pikachu in Ultimate, some dropped this fanon entirely, while others decided it was simply a new Pichu.
One popular theory for the original Pikachu as of Ultimate is that it moved to Alola and evolved into the Alolan Raichu that appears out of a Poké Ball occasionally.
It's often believed that the crowd cheer for Ganondorf is him possessing the crowd to cheer for him, since the chant for Ganondorf often sounds rather lifeless and dreary.
Some believe that the reason why Ganondorf refuses to use the bladed weapons and/or magic blasts that he's famous for in his home series is that he's so arrogant that he uses his bare hands just to give himself a challenge.
In addition to this, many people often portray Ganondorf as a Captain Falcon fanboy or as the Captain's father as a reason to explain why their moves are the same. The web animationGanon Knows Best likely helped popularize the latter, though its assertation that Falcon is Ganondorf's kid via Nana (and goes by the full name of "Captain Popo Falcon") is less widespread.
Pokémon Trainer is named "Red" in fan works, just like the character he is based on in the Pokémon games. Likewise, when Ultimate introduced a female Pokémon Trainer based on the playable female Trainer from FireRed/LeafGreen, fan works named her either "Leaf" or "Green".
Lucas is often portrayed as a Shrinking Violet and more emotionally weak than Ness. In their home games, their personalities are very similar and they go through the same Character Development. The main difference is that Lucas has had a lot of worse things happen to him in his game. Lucas was a Perpetual Frowner in Brawl and Ness did save him in The Subspace Emissary, but he eventually becomes more confident later on in the story, which carried on to for 3DS/Wii U and Ultimate.
The final boss of The Subspace Emissary, Tabuu, is commonly theorized to be Master Hand's Abusive Parent. If Master Hand symbolizes the imagination of a child playing with his toys, Tabuu would represent an authority figure in the child's life stifling his imagination by demanding he stop playing with his toys, i.e. making them taboo (in-universe, this would explain him turning the Smash characters into lifeless trophies).
Many fans take a fan comic showcasing how the Duck Hunt Dog came to join the fourth game's roster (by taking care of a baby Duck after finding out his master killed its parents) as fanon, depicting the two as True Companions.
A common fandom theory is that Captain N is the hunter in the Duck Hunt trio.
Also, to keep in the spirit of Banjo-Kazooie and Ambiguous Gendernote despite the duck clearly having a male's colorful plume, especially with Banjo & Kazooie's addition in Ultimate, some fans like to think of the duck as being female, just like Kazooie.
It's common for fan art to depict Bayonetta as Joker's mother, largely due to their similar colour schemes and fighting styles. After Kazuya was announced, he was added to the "family" due to his devil motif fitting in with that of Joker's Persona Arsène.
When it comes to drawing art of the fighters from Ultimate's first Fighters Pass, most fans mirror the Fire Emblem fandom in preferring female Byleth, largely because she's the only female fighter in the pass (not counting Kazooie). As for previous Fire Emblem Avatars Robin and Corrin, they tend to be depicted as male and female respectively.
In fan works that depict both versions of each Fire Emblem protagonist, it's common to give one of them a nickname to help distinguish them from each other; most commonly, this amounts to having one of them get referred to by their Japanese name.
Due to both of them being reptilian antagonists formally introduced as playable characters in Ultimate, it's become common in fan art to portray Ridley and King K. Rool as friends.
Ganon Knows Best