There were complex aspects about both those wars. But they were indisputably fought between Danes and Germans. Schleswig-Holstein was a German duchy under the Danish king, not a paart of Denmark proper.

The second war, in 1864, ended with a huge loss of Danish territory to the emerging German superstate. The northern half of this territory was returned to Denmark in 1920, with the southern remaining German. This was the result of a referendum in the wake of World War 1. Everything was orderly and civilized, and the border region to this day remains a model of how to handle such things without undue upheaval - thus exactly confirming my point above.