Metal Gear
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The character Meryl Silverburgh that first appeared in Metal Gear Solid is heavily based on the character with the same name from the game Policenauts. Both have remarkably similar designs, both are involved with FOXHOUND, and both even have the same Japanese voice actor, Kyoko Terase. In Metal Gear Solid 4: Guns of the Patriots, the likeness between the two is taken even further, as Meryl wears bullet-earrings like her Policenauts-counterpart and also has an Ed and Jonathan on her team.
Contributed by G-Haven
In an interview with GameInformer, Hideo Kojima said that inconsistencies in the Metal Gear canon come from prioritizing each game's message over the series continuity:

"One thing that I try not to do is sacrifice certain things just so I can keep the original story intact. Sometimes I need to accept these inconsistencies in order to be able to achieve what I want for the story."
Contributed by CarlosTheHitman
The idea of Snake hiding within a cardboard box to avoid enemy detection was inspired by the novel Hako Otoko (Box Man) by Abe Kobo, because Hideo Kojima is a fan of Kobo's novels.

The novel centers around a man who abandons the modern world and chooses to live and observe the "mad world" of Tokyo from within a "waterproof world", i.e the cardboard box he carries on his back.
Contributed by MightyKombat
In Metal Gear Solid, Master Miller is depicted with blond hair and aviator sunglasses. However, Miller in Metal Gear 2: Solid Snake is depicted as an Asian-looking man with black hair. Revisions of Metal Gear 2 from the 2004 mobile phone and games onward depict Miller in his new design.
Contributed by SOGESNAKE
The Metal Gear Solid main theme used in the first MGS, Sons of Liberty, The Twin Snakes and partially in Snake Eater was removed from Portable Ops shortly before its release because of a potential plagiarism.

Someone on the Internet found a song called "The Winter Road" by the Russian composer Georgy Sviridov which strongly resembled the theme composed by TAPPY.

When Hideo Kojima heard it and became aware of the similarities between the two tracks, he decided to not use the theme again in any future titles in order to evade legal issues.
Contributed by Gallego13
There is a recurring character within the Metal Gear Solid series called "Johnny". In each game, Snake will come across a character named Johnny, never being the same person but rather a member of a long line of Johnnys from the same family. In each instance, Johnny can suffer from severe diarrhea.

In the first Metal Gear Solid, Johnny is encountered twice. First he is found passed out and naked by the cells in which Maryl escapes, and later he guards Snake's cell before he is tortured.

In Metal Gear Solid 3, Johnny reveals a lot of back story to Snake while guarding his cell, giving insight into his family's tradition of naming every male born into the family "Johnny".

In Metal Gear Solid 4, Johnny is found suffering from diarrhea in a metal barrel, and is constantly suffering from bowel trouble throughout the story. This time, Johnny plays a protagonist, and a major role in the game's story as well.
Contributed by Dazz
There is a Yu-Gi-Oh! card which greatly resembles Snake from Metal Gear Solid.

The card is titled "Tactical Espionage Expert" and depicts a man with brown hair and a gun hiding behind a corner, dressed similarly to Snake.
Contributed by Psychospacecow