Before it was a fruit, "Tangerine" meant a place. The word began as an adjective meaning "from or relating to Tangier," the Moroccan port on the Strait of Gibraltar.
Etymonline dates that adjective to 1710 and says it probably came from Spanish tangerino. Wikipedia notes that the Oxford English Dictionary cites it from Addison's The Tatler in the same year. At that point it had nothing to do with citrus.
The fruit borrowed the label because of a shipping route. The fruit was first imported to Britain by way of Tangier, so it was sold as the "tangerine orange," a phrase recorded in 1820. By 1842 the orange had dropped away and "tangerine" stood alone as the fruit's name. The color name followed in 1899.
So the word records where British buyers got the fruit, not where it grew first. Botanically, Wikipedia describes the tangerine as either a variety of the mandarin orange or a close relative of it. Some analyses treat it as a mandarin hybrid with a little pomelo in its ancestry.
The name crossed the Atlantic with the fruit. Wikipedia credits a Major Atway of Palatka, Florida, with first growing tangerines as a distinct American crop. Atway was said to have imported them from Morocco. In 1843 he sold his groves to N. H. Moragne, and the variety became the Moragne tangerine. A seedling of it, the Dancy, became one of the oldest and most popular American varieties.
The port's own name has a myth attached. Its Latin form was Tinge, and an old tradition linked it to Tingis, a daughter of Atlas. Etymonline calls that probably a folk etymology and points instead to a Semitic word meaning "harbor." In English the city is often written "Tangiers," on the model of Algiers.
The fruit then lent its name to a newer one. In 1904 a cross between a tangerine and a pomelo was called a tangelo, built from the two fruits' names.