Most likely, either name would indicate someone who moved from one region to another (which was quite common). If you lived in Brooklyn, and moved to Chicago, everyone in Chicago would know you as “the Brooklyner”. A person with a geographic name of origin would indicate the person came from the place and settled in a place where such an indicator would be of use in identifying them.
You should also know that “Ashkenazi” sometimes refers specificially to Germany, so a yekke who moved to Eastern Europe might be known as “Ashkenazi”. Based on such evidence as family records, gene flow, etc., many people moved between regions.