Many languages took the Asian words for tea because we discovered it from them. Just like chai in English is a form of cha when in reality your "Chai tea latte" really means tea tea with milk (latte being the Italian word for milk).
You can actually tell which way tea came to different parts of the world by their word for it. Chai as used in portuguese, russian and chinese came via india, while the word tea as used in german, spanish, french and english came via indonesia I think.
You can also see Portuguese influence in Korean and Japanese thanks to missionary presence some hundreds of years ago.