Donkey Kongs' identities

Donkey Kong Country introduced Cranky Kong as the elderly form of Donkey Kong from the Donkey Kong arcade game. This has led to games and official statements providing conflicting information on the identities and familial relationships of Donkey Kong and Donkey Kong Jr..

Cranky Kong is Donkey Kong's grandfather
The interpretation that Cranky Kong is the grandfather of the modern-day Donkey Kong was first used in the SNES Donkey Kong Country games. It seems to be the current position as of Donkey Kong Country Returns, both through in-game lines, and in the instruction booklet:

"Donkey Kong’s white-bearded, grouchy grandfather, Cranky Kong, has set up shop in each world to sell items to aid Donkey Kong."

This reading was also used in Snake's codec conversation in Super Smash Bros. Brawl:

"The Donkey Kong who fought that epic battle with Mario was this guy's grandfather."

This suggests that Donkey Kong Jr. is Cranky Kong's son and Donkey Kong's father. One section of Playing With Super Power: Nintendo Super NES Classics states that the current Donkey Kong "is actually Donkey Kong Jr.'s son".

Cranky Kong is Donkey Kong's father
This relationship shifted in the Nintendo 64 era, but seems to have been reverted since. In Donkey Kong 64, Cranky Kong consistently addresses Donkey Kong as "son". This suggests that Donkey Kong is a grown-up Donkey Kong Jr., which will be detailed in greater detail in a later section.

Other promotional material from this time period has allegedly taken the same stance, but it seems that these instances have not been archived properly.

The current Donkey Kong is the arcade Donkey Kong
Some games claim that the present-day Donkey Kong is the original one, implying that Donkey Kong Jr. is his son and leaving Cranky's status unclear. For instance, his bios in Super Smash Bros. and Super Smash Bros. Melee respectively state:

"Donkey Kong and Mario started out as arch-rivals, but they've patched things up in recent years. These days Donkey Kong spends his time searching the jungle for bananas instead of kidnapping beautiful maidens."

"While he now prefers the laid-back jungle lifestyle to construction site mischief, DK is often forced back into action by the Kremling Krew."

Mario vs. Donkey Kong also advertises that "For the first time in ages, Mario squares off against his original foe!" However, its sequel's instruction booklet claims that "He instantly fell in love with Pauline as soon as he saw her at the theme park’s opening ceremony," which seems to contradict this theory by removing the Donkey Kong arcade game from the backstory. As mentioned, Super Smash Bros. Brawl marks a similar change in stance for its series.

Interestingly, these games all use Donkey Kong's modern appearance introduced in Donkey Kong Country, despite it specifically being designed for the younger relative of the arcade character.

The current Donkey Kong is Donkey Kong Jr.
Another section of Playing With Super Power: Nintendo Super NES Classics claims that Donkey Kong "is, in fact, Donkey Kong Jr., all grown up." This would presumably make him an older version of Baby Donkey Kong.

However, the Prima Games Twitter account has clarified that this passage is a metaphor for the character's redesign in Donkey Kong Country, and thus they are not literally meant to be taken as the same character.

This interpretation is arguably also contradicted by Donkey Kong Jr. being playable alongside Donkey Kong in Mario Tennis for the Nintendo 64; and his cameoing in the audience of Waluigi Stadium in Mario Kart: Double Dash!!, another game where Donkey Kong is playable. This would further suggest that Donkey Kong Jr. is younger than the modern Donkey Kong. However, both games feature Baby Mario as a playable character alongside Mario, allowing this to be resolved and contended via implied time travel or cloning.

Further potential evidence against this reading is that in Yoshi's Island DS and Mario Super Sluggers, Baby Donkey Kong wears a bib resembling the "DK" tie. This is mostly associated with the modern Donkey Kong, not Donkey Kong Jr., who wears a white leotard with the letter "J".

According to GameXplain, the paradox of Donkey Kong Jr. is that he is meant to be Donkey Kong's father but will always be presented in a younger form, due to a lack of modern relevance making him most recognizable that way.

Baby Donkey Kong is not the baby form of the current Donkey Kong
Despite Baby Donkey Kong's appearance and bib most closely matching the current Donkey Kong, GameXplain claims that it doesn't make sense that Baby Mario is both the same age as Baby Donkey Kong, and grows up to fight the arcade Donkey Kong.

A common alternative fan theory, which attempts to resolve the general timeline of Cranky Kong's aging, claims that Jumpman and Mario are separate characters. Under this theory, Jumpman is usually interpreted as an older relative to Mario, like a father or grandfather, as Cranky Kong is to Donkey Kong. However, Pauline's dialogue in Super Mario Odyssey suggests that she and Mario were the ones involved in the events of the Donkey Kong arcade game.

Diddy Kong is Donkey Kong Jr.
In Donkey Kong Country and Mario Kart: Double Dash!!, Diddy Kong took on playable roles that were meant for a redesigned Donkey Kong Jr. earlier in development.

Although this is not canon to the final versions of the games, these plans imply that Donkey Kong Jr. would be younger than the current Donkey Kong even if they appeared in a game without time travel elements. This would call into question Donkey Kong Jr.'s position as Donkey Kong's father, or otherwise as the son of a grandfatherly Cranky Kong.

Junior (II)'s identity
A pink palette swap of Donkey Kong Jr., named Junior (II), appears as the second playable character in most versions of Donkey Kong Jr. Math. What role and familial status Junior (II) is intended to have in the Donkey Kong canon, if any, is up for interpretation.

Notably, Junior (II) was the focus of Super Mario Wiki's 2019 April Fools Day prank. Rebranded as We Are Pink Donkey Kong Jr., the wiki joked that Cranky Kong adopted Junior (II) and painted him brown to hide Donkey Kong Jr.'s disappearance from the other Kongs. The prank further claimed that Cranky Kong and Donkey Kong have no official familial relation as a result.