Twitch's design closely resembles that of the Skaven, a devious and warlike race of anthropomorphic rats from the Warhammer Fantasy franchise. The Medieval skin reflects this similarity more than his original skin.
Deadly Venom displays an "X" symbol upon reaching 6 stacks. This is most likely a reference about how infected houses during the Bubonic Plague would be marked with an "X" on the door.
Twitch's title is likely a reference to the Bubonic Plague, which killed millions in Europe during Medieval times, and was transmitted by rats or more specifically, their fleas.
Upon receiving a visual update, Twitch's ultimate was renamed from Spray and Pray to Rat-Ta-Tat-Tat. While rat-a-tat-tat is a widely known onomatopoeia for gunfire, many believe this is in reference to the Pokémon Rattata, whose name is often mispronounced as Ratatatta (rat-ta-tat-ta).
The quote "Boop!" is a reference to the common internet image macro of the same name, usually involving an animal being touched lightly on the nose.
Twitch's special taunts for Singed reference a common and unofficial "rule" of the game: Don't chase Singed. Because of his unique playstyle and skillset leaving a trail of poison behind him, Singed typically does well against newer players unfamiliar with this rule.
Twitch's taunt "I hear you're trash! Bow before your king!" could be a reference to the professional player Yiliang "Doublelift" Peng, who is known for smack talk and specifically the phrase "Everyone else is trash".
Twitch's quote "I demand a sandwich!" is a reference to a popular meme quote by Carlton from the Fresh Prince of Bel-Air, "Would you make me a sandwich?".
It was conceived by the 2 people that first hit 350 friend referrals: Grez and Greyrir Wolfwood. Grez picked the champion that would receive the skin, and Greyrir Wolfwood designed a "Medieval Twitch, with a fleur-de-lis on his cape, and a pot on his head". The idea was given to the Riot artists and Medieval Twitch was released a few months later.
Medieval Twitch holds the distinction for being the rarest skin in the game.