Quixotic is named after a man named after a piece of armor. And English speakers pronounce it in a way no Spaniard ever did.
The man is Don Quixote, hero of Miguel de Cervantes's novel, published in two parts in 1605 and 1615. The book finally names him Alonso Quixano, a country gentleman nearing fifty in La Mancha. He reads so many tales of chivalry that he decides to become a knight-errant himself. In the best-known scene he charges at windmills, convinced they are giants.
The name he gives himself is a joke. In Spanish, quijote is a cuisse, the plate armor that protects the thigh. Etymonline traces it to Latin coxa, "hip." The ending adds to the comedy. The Spanish suffix -ote makes things bigger, often in a grotesque way. Wikipedia reads the name as a mock "Great Quijano," built on a leg guard.
English adopted the character quickly. Thomas Shelton's translation of the first part appeared in 1612, while Cervantes was still alive. According to Wikipedia, "quixote" was already used for an impractical dreamer in a 1644 work by John Cleveland. The adjective followed. Merriam-Webster dates quixotic to 1718. Etymonline gives 1791. Either way, the word has meant "extravagantly idealistic" for about three centuries.
Then there is the sound. In the Spanish of Cervantes's time, the letter x stood for a "sh" sound. Spanish later shifted that sound to the throaty "kh" heard in modern Quijote. English ignored both. It read the x as an English x, giving "kwik-SOT-ik." That is why many English speakers say the hero's name the Spanish way but the adjective the English way.
The book also gave English a second idiom. To tilt at windmills is to fight imaginary enemies, straight from that scene. Spanish keeps a neat pair that English lacks. Quijotesco means quixotic, while sanchopancesco, after the squire Sancho Panza, describes someone practical and unidealistic.