Nobody ordering a cappuccino is thinking about monks, but that is where the name comes from.
The Capuchins are a Franciscan order, a branch of the friars that follow St. Francis. They are named for the long, pointed hood on their brown habit. Etymonline traces the English word through French to Italian cappuccio, "hood." So a Capuchin is, literally, a person with a hood, and a cappuccino is a "little hood."
The drink borrowed the name for its color. Merriam-Webster says the coffee is named for the likeness of its color to a Capuchin's habit. Etymonline says much the same, but puts it as the brown of the friars' hoods.
The route from habit to cup probably ran through Vienna. Britannica says the earliest record of the drink is the Kapuziner, served in Viennese coffeehouses in the 1700s. Accounts differ on the recipe. Some say whipped cream, others a few drops of cream, but the pale brown color is what recalled the friars.
Italy then made it a different drink. Britannica says the Italian cappuccino appears in the early twentieth century, after the espresso machine was invented. The foam on top is a modern, milk-based echo of a friar's hood."
Dictionaries disagree on when the word first appears in English. Merriam-Webster gives 1893, while Etymonline dates it to 1948. They may be counting different senses, so the safest claim is only that the coffee sense was in English by the middle of the twentieth century.
The friars lent their name elsewhere too. The Capuchin monkey is named for the cowl-like cap of hair on its head, and Etymonline dates that use to 1785.