Europe has had a common history for thousands of years. Constantly forming, splitting and reforming. The Hellenistic Greeks were all over the Mediterranean, the Roman empire was all over the place, these England was a subsidiary of the kingdom of France, then not, then Ireland of England. Netherlands was part of Spain, Belgium was part of France. Lots of western Poland was in Germany, Austria was part of Germany, Nice (now in France) was part of Italy.
We're one big family who've been all over each other for thousands of years.
Technically Nice was never a part of Italy. Nice was most of the time part of the Duchy of Savoy (later called the Kingdom of Sardinia when the Duke of Savoy also became the king of Sardinia). Nice became French in 1860 while Italy was formed in 1861.
What used to be the Duchy of Savoy is now split between Italy and France.
When talking about history there are very few "correct" and "right" answers, since one would have to first define what one means by "Germany" and "Austria".
I believe the Hapsburg monarchy controlled a larger portion of it than the Prussians, so it technically it's closer to Germany having been part of Austria than the other way around.
Of course it is even more accurate to say neither is true, since the holy roman empire was more of a loose confederation than a country, i.e. closer to something like the EU than to Germany or Austria.
And what about East Fracia, or even the Carolingian Empire itself? Austria & Germany together.
That's the thing with this history stuff, it's rarely as clear cut, nor are today's examples (e.g. modern Germany, EU) totally relevant for examples of the past.
We're one big family who've been all over each other for thousands of years.