(А general term for 'machine' not just a 'car/automobile' 'Автомашини'? Sewing machines? ATO Tanks? Hydraulic machines? Farm combines? Etc.... There are SO many specific 'machines'! 'Автомашина' is the specific term for 'automobile'(a.k.a. 'car'.)
You are right, "машина" can mean any machine in Ukrainian, but it is also a common word for "car." So, if you are asked "У вас є машина?" (and the context does not suggest something different) then "car" is meant.