German was an ethnicity long before it denoted someone who is from the nation state of Germany. For centuries, the signifier that denoted someone as German was the language they spoke, not the state under whose jurisdiction they lived or were born. In the age of Mozart, most people would not have identified as being a member of a particular nation state, but rather as a member of a particular confession, a member of a particular ethno-linguistic group, or perhaps a member of a particular profession or guild.

The question of whether Mozart, for example, was Austrian or German is anachronistic, in the sense that the nation states of Germany and Austria did not exist at that time. Rather, Mozart was Catholic, ethnically German (his dad was actually from the Bavarian city of Augsburg), and a subject of the Hapsburg Emperor (who would declare himself the Emperor of Austria about 15 years after Mozart died). Mozart is most commonly associated with the present day Austrian city of Salzburg, this place of his birth, and Vienna, the city where he spent the longest portion of his life and where he died.

Anyone who tells you that Mozart was Austrian, not German either has a political point to make (probably about Austria being distinct and separate from Germany, which is really a post-war development and simply doesn't apply to Mozart's age) or they are confused about the nature of how things worked back then.