Order "un latte" at a bar in Italy and you will get a glass of milk. The word means milk, and nothing more.
The drink's full name is caffè latte, Italian for "milk coffee," sometimes written caffè e latte, "coffee and milk." English has used the full phrase for a long time. Wikipedia reports a first English attestation in 1847, in The Cultivator of Albany, New York. The American novelist William Dean Howells later described being served a "cafè latte" on a steamship from Trieste to Venice.
The short form is recent. Etymonline dates latte on its own, meaning espresso with milk, to about 1990. English speakers dropped the coffee and kept the milk, which is exactly the half that does not identify the drink. That clipping is why the bare word works in an American café and fails in an Italian one.
Where the modern drink took shape is a separate question. Coffee with milk is often traced to 17th-century Europe, and in Italy milky coffees are often served only at breakfast. The tall espresso-based latte depended on the steam wand, added to espresso machines in 1903, which could heat and texture milk.
One California café claims the American version. Caffe Mediterraneum, opened in Berkeley in 1956, says the drink "was invented here" in the late 1950s. Its account says co-owner Lino Meiorin made a bigger drink with more steamed milk for customers who found cappuccino too strong. Daily Coffee News reports the claim without independent evidence. Even the café concedes that Seattle may have made the drink famous.
The Latin root has a big family in English. Lac, "milk," gave lactose, lactate and lactation. A related Greek word, gala, gave galaxy. The French cousin of the drink, café au lait, entered English in 1763, with lait from the same Latin lactis.