Translation:Give me the overcoat and the hat, please.
Agree. "A" is especially not needed if you said "the coat." I had, "give me the coat and hat," but Duo corrected it to "the coat and a hat," which is silly. "The coat and the hat" makes sense if he is ready to buy these after trying them on. "The coat and hat" makes sense if he checked them together coming into a night club and is now ready to leave.