Germany as a region, as a nation, if not a state, had existed for a very, very long time, starting with the Romans, who called the whole region Germania. The various Germanic states of the Holy Roman Empire were often called The Germanies, and the Kingdom of Germany was one of the number of titles of the Holy Roman Emperor.

Even speaking to things closer to 1871, the German Confederation was super-state that existed from 1815. Germany being called Germany wasn't a new concept by a long shot.

As /u/BasqueInGlory hints at, Germany wasn't a new concept. What actually happened in 1871 wasn't strictly speaking the creation of a new country, but rather, the Kaiser being crowned over the North German Confederation and then the South German Confederation as well. The development of a truly centralized country actually really didn't happen until after the First World War; until 1919 Bavaria, Saxony and Württemberg all still maintained separate standing armies!