In video games it's very common to have a second or third jump in the air, despite that not being a thing humans can do. How did this idea become prevalent in video games, and did this idea originate from somewhere other than video games?
We don't know specifically, but we can speculate.
Dragon Buster, a Namco arcade game from 1984, was the first known appearance of what we would call the vertical double jump. The interesting thing is that this game is not an action platformer where it's important to be precise crossing gaps and whatnot. It appears - this is the speculation - that this was a crutch used by the developers. They could only conceive of the character being able to jump half their own height, but they wanted to have a number of vertical obstacles. Given that the character gets kicked around in the air a lot, it could be that they wanted an extra amount of "hangtime" to be within the player's control. We currently don't know who designed Dragon Buster so we wouldn't even know who to ask.
However, Dragon Buster likely is not the popularizer of the concept. Plus, there is a secondary version of the double jump that might help illuminate the idea.
In Smurf: Rescue in Gargamel's Castle, there is a sort of secondary jump mechanic. After jumping once, in a few frame window, you can jump again and receive a higher leap with some forward momentum. This appears to be drawing the idea of acrobats who can often do a springing leap after doing a minor jump into the air for the first time. This style of double jump was seen in other games and reveals a more realistic inspiration for platformers using this fanciful idea.
Super Mario 64 has this idea of a double - and even a triple - jump. Internally at Nintendo, Mario games were referred to as "athletic games", not "platformers". The framing of this idea of being about athleticism reveals much about the things that a character like Mario does. Though it was never used as a marketing term, it seems that a lot of other developers took to this similar spirit.
The game that likely made the double jump a widespread mechanic was the 1989 arcade game Strider by Capcom. This dazzling game wasn't hugely successful in Japan, but it was very popular in the US and Europe. Strider justified the double jump by having the character do a somersaulting motion, again similar to Olympic acrobats who are trying to push their body higher in a vaulting motion.
What this did for the design was to allow a certain amount of leeway with the jump. A single jump in a game like Strider doesn't have Mario-style adjustment of position while in mid-air. That's also something that's unrealistic, but designers go with trade-offs. In this case, they opted for making something cool in the double jump to make sure players had a slim lifeline if they missed their target.
There were a few other takes on what would happen after pressing the jump button while in mid-air. Super Mario World for instance had the cape, which was copied in other games with a gliding mechanic. Kirby made the "infinite" jump a core mechanic of its player-friendly design. These all have the same goal: Give the player leeway in platforming. If the challenge is less about making precise leaps, then there's plenty to be gained by adding some variation on mid-air control or the double jump.
Sorry if this answer seems indefinite. Not many developer interviews go into the specific nature of characters' jumps. After Donkey Kong came out, jumping was just a thing video game characters could do. At the moment we just have to try and decipher intent from the games themselves. In my opinion, the various forms of double jump were about translating athletic concepts into gameplay terms while making them friendly to players. Realism - especially from Japanese designers - takes a back seat next to fun.
Sources:
The Untold History of Japanese Game Developers: Volume 1 by John Szczepaniak.