The Kong Family Legacy
The Kong family is heavily featured in The Super Mario Bros Movie, though the details of how these characters are related aren't explained within the film. Several Kongs from the Super Mario video games were thrown into the Illumination movie, with a couple playing significant roles within the plot. While this meant some exciting cameos, it has led to confusion about the Kong family tree and the general lore of Jungle Kingdom. Thankfully, past Donkey Kong video games have the answers.
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The character of Donkey Kong was first introduced in the Donkey Kong arcade game in 1981, which also saw the introduction of the hero, Jumpman, who would later become Mario. From there, the Donkey Kong franchise grew independently from the Super Mario franchise, and with each installment came the introduction of new Kong characters—Cranky Kong, Donkey Kong, Diddy Kong, and many more. Simply having the 'last name' of Kong doesn't necessarily indicate that these apes are related since the word refers to their species rather than a family name. Still, many of the Kong characters, especially those seen in The Super Mario Bros Movie, share a connection.
Screenshot from the classic Donkey Kong arcade game
Cranky Kong: The Patriarch of the Jungle Kingdom
In The Super Mario Bros Movie, Cranky Kong is the king of the Jungle Kingdom and the father of Donkey Kong. However, this is a bit of a shift from canon. Traditionally, Cranky Kong is DK's grandfather, and his real name is actually Donkey Kong as well. He is the original character of this name and is meant to have been the villainous version of Donkey Kong seen in the 1981 arcade game. When he grew old and irritable, he began calling himself Cranky Kong. Since the Mario movie saw Mario and Standard DK fight it out in an arena that resembled the stage of the Donkey Kong game, it can be concluded that Cranky King's canon story was disregarded.
Donkey Kong and Cranky Kong celebrate together
Cranky Kong has a wife in Donkey Kong canon named Wrinkly Kong, but she doesn't appear in The Super Mario Bros Movie.
Crankys Cabin Donkey Kong Country
The Kong Lineage: From Donkey Kong to Kiddy Kong
Modern-day Donkey Kong, who was voiced by Seth Rogan in The Super Mario Bros Movie, became the primary version of the character in Donkey Kong Country in 1994. In canon, his father is Donkey Kong Jr., who appeared in the arcade game of the same name in 1982. This means that the version of DK that has appeared in every game since 1994, including The Super Mario Bros Movie, is technically Donkey Kong III. However, going by the Mario movie, he himself would be junior.
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Diddy Kong is seen in the arena audience in The Super Mario Bros Movie, cheering on Donkey Kong from the arena audience. The character was first introduced alongside modern Donkey Kong in Donkey Kong Country and is described as DK's best friend and nephew. However, it's never clear precisely who Diddy Kong's parents are. It can be assumed that Donkey Kong has either a brother or sister, but it's possible that 'nephew' isn't meant literally. Regardless, in every interaction of the character, Diddy Kong looks up to Donkey Kong, which makes his enthusiastic cheering in the Mario movie rather fitting.
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Dixie Kong is seen next to Diddy Kong in The Super Mario Bros Movie, but her part is silent in the film. She first appeared in Donkey Kong Country 2: Diddy's Kong Quest and has always been described as Diddy Kong's love interest and girlfriend. Therefore, she isn't related to Donkey Kong in any way, though some of her other family members, cousins Chunky Kong and Kiddy Kong, did appear in the Mario movie also.
Dixie Kong Donkey Kong Tropical Freeze
Chunky Kong gets a nice close-up cameo in The Super Mario Bros Movie, where he is among the audience of Donkey Kong and Mario's big fight. Chunky isn't related to DK in any way, but he is said to be Dixie's cousin. He first appeared in Donkey Kong 64, where he was an unlockable playable character. Chunky is the strongest of all the Kongs but is also something of a gentle giant, preferring butterflies, dancing, and playing musical instruments. Since Donkey Kong 64, he has only appeared in cameos in a few other DK games, which makes his appearance in the Mario movie all the more rewarding.
Chunky Kong in the Mario Movie
Kiddy Kong is the toddler of the Kong family, which makes his cameo driving a kart on Rainbow Road in The Super Mario Bros Movie delightfully ludicrous. He's the younger brother to Chunky Kong, and cousin to Dixie Kong. Kiddy first appeared in Donkey Kong Country 3: Dixie Kong's Double Trouble! in 1996, where he is seen to be surprisingly strong for his young age. Since then, he has only had a significant appearance in one other game, Donkey Kong Land III (1997), though he has had minor cameos in other Donkey Kong and Super Smash Bros titles.
Kiddy raises arms in Donkey Kong Country 3
The Mysterious Kongs: Swanky, Funky, and More
Swanky Kong is another memorable Kong seen in the audience during DK's fight with Mario, noticeable because of his 'swanky' appearance. The character has no known family connection to any of the other characters, though his close resemblance to Donkey Kong has led some to theorize that he is a brother. This means he could be Diddy Kong's father or uncle. Swanky Kong first appeared in Donkey Kong Country 2: Diddy's Kong Quest in 1995 as a minigame host and repeated the same role in Donkey Kong Country 3: Dixie Kong's Double Trouble! (2005 port), but he hasn't appeared in a game since.
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Another character seen driving a kart on Rainbow Road is Funky Kong, who is recognizable for his bandana and sunglasses. Though his character cameo was rather small in The Super Mario Bros Movie, he has appeared in far more Donkey Kong and Mario titles than most of the other Kongs in the film. Like Swanky, he isn't confirmed to be related to Donkey Kong, but he is close enough in appearance that he could potentially be a cousin or sibling. Typically, Funky Kong is regarded as a friend to the family and is known to offer a helping hand (often in the form of transportation) in sticky situations.
Funky Kong from Donkey Kong Country series in a cropped picture.