I've been researching this topic vigorously for a few hours now, haha.
It's actually quite easy to see why Hitler considered himself a German, despite the fact that he was born an Austrian citizen since he was born in Austria-Hungary, not the German Empire.
Up until 1866 Austria was considered to be as much a part of Germany as Bavaria, Prussia, Saxony, and all of other independent German states. In fact, the Austrians (The Habsburgs) had ruled Germany (The Holy Roman Empire of the German Nation (HRE)) for six hundred years. When the HRE was dissolved in 1806, the German Confederation was formed and the two biggest German states - Austria and Prussia - became rivals and the Austrians and Prussians had different ideas about how Germany should be unified as a nation-state.
In the 1800s the Austrian Empire ruled a lot of non-German territory and they were unwilling to give up the territory so the Prussians didn't want to include the Austrians. The Austrians favoured a Greater Germany to include Austria and the Austrians and the Prussians favoured a Little Germany without Austria and the Austrians. Things came to a head in 1866 during the German war and as it turned out the Prussians defeated the Austrians which subsequently excluded both Austria and the Austrians from the newly formed German Empire in 1871. The Bavarians and Bavaria only joined the German Empire because the Bavarians helped the Prussians defeat the French in the Franco-Prussian War in 1870.
A few facts I have discovered:
The Prussians weren't even originally a Germanic tribes. They were a Baltic tribe who were later Germanised.
The Austrians don't speak German because of a mistake - it is because they are Germans.
The name of Austria in German - Österreich - refers to the 'eastern realm' when Austria was formed from the Margraviate of Austria, a borderland of the Duchy of Bavaria in 976. That explains why so many cities and towns have changed from Austrian rule to Bavarian rule thoroughout history, including Hitler's birthplace, Braunau am Inn. No wonder Hitler described the town as "Bavarian by blood" in Mein Kampf.
In 1918, after the end of WW1, Austria was a rump-state and renamed itself 'The Republic of German-Austria' and declared itself to be a part of the German Republic, but the victors of WW1 forbid the name and the union between the two countries.
In 1938 when Hitler annexed Austria it made his popularity was greater than ever before.
I understand why it's a touchy subject for Austrians today, but people shouldn't ignore facts.
Mozart also considered himself to be a German and his birthplace, Salzburg, was not a part of Austria when he was born.
Up until 1945, Austrians were seen as just another type of Germans like Bavarians, Prussians, etc.
If history had turned out differently then the Austrians would have unified Germany and excluded Prussia, would people then be saying Prussians were not Germans? 
Although Hitler only became a German citizen in 1932, less than a year before he became the Chancellor of Germany, it is ridiculous to claim that he was 'not German'.
The British historian Alan Bullock in his book "Hitler and Stalin: Parallel Lives" wrote:
"Hitler, of course, was a German, but he was born a subject of the Habsburg Empire, where Germans had played the leading for centuries. However, with Bismarck's creation in the 1860s of a German Empire based on Prussia, from which the Austrian Germans were excluded, the latter found themselves forced to defend their historic claim to rule against the growing demands for equality of the Czechs and the other "subject peoples"."