Apparently this is one of the times that nusach sfard is correct in keeping the original minhag. The Ramma 581 siff one says to blow shofar in the erev. The Mishna Berurah when tagging on the minhag to say lidovid mentions that erev is after mincha. The Chayei Adam and Aruch Hashulchan say the same. The Igros Moshe in O'ch part 4 siman 21 ois 5 explains the minhag of blowing shofar in the evening had to have been during the day because we can't blow shofar from the beginning of the night untill chatzos due to it being a time of din. He explains that erev means after mincha and brings a proof for this. Or he says it was actually after Maariv but only if they davened mib'od yom, while it was still daytime. He goes on to explain why we only blow once in the morning.
I will add that being that the shofar blowing and saying lidovid are put together in this sugya, than its possible the ashkenazi minhag started when everyone was davening maariv before nightfall and continued even when people started davening after nightfall.