I wonder if the same feelings held for Germans on the Confederate side when they fought against Unionist Germans many of whom, as you point out, were off the boat immigrants. Somewhere in my digital library I have an article from The Virginia Historical Society which recites a story of a Confederate officer sent from Richmond to recruit the German yeomanry of the Shenandoah Valley. The story says he mounted a wagon to harangue the crowd when a voice interrupted saying "it is okay, you can speak English to us."
The Germans who had pioneered the Shenandoah Valley to settlement beginning in 1726 spoke German as well as English at the time of the Civil War and continued to do so until the government put stop to the practice in the schools and elsewhere at the time of the first world war.
Many of the regiments who fought under the Stars and Bars were solidly German and marched with Stonewall Jackson in his famous Valley campaign of 1862 and can rightfully claim to have participated in virtually every major engagement in the Eastern theater from The Seven Days until Appomattox.
It would be interesting to know if there was any regret on either side at killing other Germans who had become very American or who were rapidly in the process of doing so.