The Milky Way is also called by the same name in other languages, of cause with the respective language change - Via Latea in Spanish for one. So with the 'original' Greek term, it could be argued that it is called the Milky Way Galaxy. Professional astronomers also call it such. I guess somewhat by default,
BUT! The Greeks were not the first to name what we see in the sky as The Milky Way! There are Australian Aboriginal words that give a 'name' or description to it too. Aboriginal astronomy is very rich in nomenclature. Everything we see in the sky has many, many different names, all according to the different languages spoken. Some saw it as a river, others as a rift or gash in the sky. Some saw it as the place where spirits of the deceased went to, to then look down over Earth, forever then having a direct link with the living (the spirits were taken up to the sky in a canoe. Meteors we see is the canoe returning to Earth). Nocturnal navigation also demanded a very precise set of nomenclature - cooler to travers long distances at night than during the day, something our very own SAS troops do today too and not just for cover.
Other old cultures also name the MW. Those cultures isolated from European influence all did. The Aztecs, Maya and Inca from North and South America all did as an example.
While the Greek name is regarded as 'accepted', it by no means is the only term, or the first.