Hitler was ethnically German. The part of the old Austro-Hungarian empire that became the modern country of Austria after World War I (which was the part of the Austro-Hungarian empire Hitler came from) was the German-speaking part. Up through the nineteenth century Germany was a geographical region that included lots of countries, most quite small. In the nineteenth century the two largest German-speaking countries were Prussia and Austria. Prussia united most of the other German countries (including the Kingdom of Bavaria) under its leadership and formed the German Empire in 1871. While the new German Empire did not include Austria, the German-speaking Austrians were still considered culturally and ethnically German.

He even served in a Bavarian regiment during WW1 and that was almost certainly a mistake on behalf of the Bavarian authorities.

It wasn't a mistake. The Bavarian authorities knew he was an immigrant from Austria. When the war broke out, Hitler submitted a request to the throne (King Ludwig III of Bavaria) for permission to volunteer for the Bavarian army, and King Ludwig gave permission (John Toland, Adolf Hitler, 1977 paperback, page 78).