The difficulty with such a question is that ethnicity in large is a social construct of modernity. Meaning that before - slightly simplified - before the 19th century, when people said they were German or Austrian, they meant something entirely different from the concept that is described today with these terms.

Before the advent of nationalism, being German or Austrian was in some ways used but from what we can tell as historians, it took a back seat to your more local affiliation. Basically, whether German or Austrian or subject to Habsburg or someoneelse was only secondary to being Tyrolean, Caranthian etc. etc.

It was only in the 19th century when nationalism arose in Europe and the Habsburg empire, in part due to the circumstances of the Napoleonic wars that within the Habsburg empire, the German-speaking Austrians started to begin thinking of themselves as ethnic Germans - or alternatively as ethnic Austrians. Constructing a past going back hundreds of years, arguments why they were German or not raged within the empire, mostly along political lines and especially when it came to if and how to unify Germany as a state. Proponents of a so-called "Kleindeutsche Lösung" would argue that Austrians and Germans were distinctly separate and therefore there should be no participation in a unified Germany. The arguments at some point became moot but lingers on up until today.

Just how constructed the whole thing is can be seen in the period after 1945 and up until today. While in the first republic the majority of the population would have happily claimed that Austrians were ethnic Germans, post 1945 Austrian nationhood and ethnicity found a new audience and was rapidly spread, especially by conservatives and communists. Measures ranging from celebrating 950 years of Austria in 1946 to renaming German in schools as "Unterrichtssprache" and so were measures taken by the governement in order to make people more Austrian.

And it worked:

In 1964 47% of all Austrians believed Austria to be a nation while 23% said that Austria was beginning to feel like a nation and 15% claimed that Austria was not a nation. Two years later a more specific survey showed that 35% of all Austrians agreed with the statement that Austria is a completely independent nation, 29% agreed that Austria is an independent nation despite belonging to a German language and culture area while approximately 20% stated that Austria was not a nation. (Bruckmüller, pp. 383-384).

In contrast, in 1993 80% of all Austrians believed Austria to be a nation and only 6% agreeing that Austria is no nation but also 61% of Austrians felt very and 31% felt quite proud to be Austrian in the same year. (ibidem.)

So, accepting ethnicity as the invention of modernity it is, there were several times, when Austrian identity was quite different from German identity.

Sources:

• William T. Bluhm: Nationa Building: The Case of Austria. In: Polity, Vol. 1, No. 2 (Winter 1968), pp. 149-177.

• Ernst Bruckmüller: Die Entwicklung des Österreichbewusstseins. In: Robert Kriechbaumer (ed.): Österreichische Nationalgeschichte nach 1945. Die Spiegel der Erinnerung: Die Sicht von innen, Wien/Köln/Weimar 1998, pp. 369-396.

• Ernst BruckmÜller: Österreichbewußtsein im Wandel, Wien 1994.

• Fritz Fellner: The Problem oft he Austrian Nation after 1945. In: The Journal of Modern History, Vol. 60, No. 2 (June 1988), pp. 264-289.

• Margarete Lamb-Faffelberger: Beyond „The Sound of Music“: The Quest for Cultural Identity in Modern Austria. In: The German Quarterly, Vol. 76, No. 3 (Summer 2003), pp. 289-299.

• Harry Ritter: Austria and the struggle for Austrian identity. In: German Studies Review, Vol. 15 (Winter 1992), pp. 111-129.

• Gerald Stourzh: Vom Reich zur Republik. Studien zum Österreichbewusstsein im 20. Jahrhundert, Wien 1990.

• Edward Timms: National Memory and the „Austrian Idea“ from Metternich to Waldheim. In: The Modern Language Review, Vol. 86, No. 4 (October 1991), pp.898-910.