Lippe-Detmold was an alternative name for the Grafschaft, later Fürstentum, later Freistaat Lippe used to distinguish it from holdings of other branches of the ruling house, especially Schaumburg-Lippe. Lippe lost it's independence in 1948 and was incorporated into Nordrhein-Westfalen where it's territory now roughly approximates the Kreis (county) Lippe in the Regierungsbezirk (next higher administrative unit) Detmold. The city of Detmold is also the administrative seat for the Kreis Lippe. Using Lippe-Detmold does not mean somebody was from the city of Detmold but from the land.
It is not common to use the county in giving locations in Germany, Detmold would just be listed as "Detmold, Nordrhein-Westfalen". If there is a need to distinguish between towns with the same name most commonly some natural feature is used, for example Frankfurt am Main and Frankfurt an der Oder. Very rarely, when there is no widely known natural feature is available, the Kreis is used, for example there are over a dozen Weißensteins in Germany, one of which not having any other distinguishing feature used to be listed as Weißenstein, Kreis Göppingen.