The word coach has two distinct meanings: “athletic trainer” and a “carriage” of some kind. The former meaning came along much more recently; it was coined by college students in the late nineteenth century as a metaphor for one who "carried" student athletes to success (so yes, the words are related). The "carriage" definition came into English in the sixteenth century from French coche, after which it followed an east-to-west, going backward to German kutsche and Hungarian kocsi. All the definitions of these words followed a semantic structure having to do with transportation, but that all changed with the etymon of kocsi. This word actually was named after the local village of Kocs, because in olden times they had many cart manufacturers down yonder, and metynomy worked its magic. Since the Kocs carts were so amazing, they also made a linguistic impact from Slavic to Romantic languages.
Adam Aleksic, a rising sophomore studying linguistics and government at Harvard University, has been described as the internet's sixth most famous etymologist. He also has disturbing interests in politics, vexillology, geography, board games, conlanging, and law.