Latte doesn't actually have coffee in it? It means 'milk'. It's a style of coffee, but if you just order latte without any shots of espresso, you get a cup of steamed milk. Of course, most places default to one shot, but literally it's just hot milk with something else in it.
Milk tea doesn't use the 'latte' word because it's from Asia, not Italy; not because a Matcha Latte wouldn't make sense (it would).
...And apologies for being waaay to geeky about coffee house drinks... x-x One of the first bots I wrote was an espresso stand. And well, I grew up outside of Seattle before Starbucks had edited the legs from their siren logo.