I think you meant "near my car"? This is close, but not completely the same. It would be "in der Nähe meines Autos". The difference is, "near" only says it's rather close, but it need not be adjacent. So "near" could well be down the street, round the corner. "next to" means immediately adjacent, and so does "neben".
I'm a native English speaker, and auto can also be used as short for automobile. I agree it's not common, and sounds hoity toity, but if someone said "Your auto is next to mine" I would understand them.
I would not think they were talking about a self-driving car. I've never heard one of those called an Auto.