We tend to refer to Austria-Hungary with its full title, but when talking about it possessively we often refer to it as Austrian and its people as Austrians.
A similar phenomenon happens with Poland-Lithuania and Bosnia and Herzegovina, which are respectively Polish and Bosnian.
Why and when did this begin to occur?
Do we?
When talking about Austria-Hungary we also talk about for instance - Hapsburg army, German bureaucracy, royal-imperial style. When talking about conflicts between for instance Czech and Hungarian, no one refers to them as Austrians (I hope).