As I bet you guessed, autó is reference to a car being self-propelled as opposed to a lovas kocsi that is pulled by a horse and other types of "kocsi" pulled by hands etc. As an interesting (at least to me) side note, kocsi literally means from Kocs which was the city where some sort of original carriage was invented. This is why some form of the word kocs shows up in a lot of languages. Even English uses coach sometimes although carriage and cart have become a bit more common.
autó is the foreign word for this vehicle, kocsi (--> coach) is of Hungarian origin https://en.wikipedia.org/wiki/Coach_%28carriage%29
My experience with Hungarian is this is not a true schwa sound, it is because it is spoken quickly in between the "és" and the "a" of autó. Slowly and carefully would make the "a" the same whether "a" or "az", which is not like any other European language "a" sound that I am familiar with.
"a" is never an [a], regarding IPA, there is only [a:] which is written á in Hungarian. "A" is indeed closer to schwa but not exactly that, it's more open and clearly more of a back vowel. This is a good reference imo https://en.wikipedia.org/wiki/Hungarian_phonology#/media/File:Hungarian_vowel_chart_with_rounded_short_a.svg