In 1806 the HRE is dissolved but the empire of Austria was instituted in 1804, in reaction to the coronation of Napoleon. The (ex)Holy roman emperor, can still be called emperor, but in a more "national" sense. Nevertheless the complete titulature of the Austrian, and then the Austro-Hungarian emperors are still a list of seperated titles, like king of Hungary, duke of slavonia etc.. The archduchy of Austria is one of those titles. By the way here is the list of the titles of Francis Ist, taken from the entry "House of Habsburg" on wikipedia: Emperor Francis I of Austria used the official full list of titles: "We, Francis the First, by the grace of God Emperor of Austria; King of Jerusalem, Hungary, Bohemia, Dalmatia, Croatia, Slavonia, Galicia and Lodomeria; Archduke of Austria; Duke of Lorraine, Salzburg, Würzburg, Franconia, Styria, Carinthia, and Carniola; Grand Duke of Cracow; Grand Prince of Transylvania; Margrave of Moravia; Duke of Sandomir, Masovia, Lublin, Upper and Lower Silesia, Auschwitz and Zator, Teschen, and Friule; Prince of Berchtesgaden and Mergentheim; Princely Count of Habsburg, Gorizia, and Gradisca and of the Tyrol; and Margrave of Upper and Lower Lusatia and Istria".

In essence, what /u/liwios said.

The Austrian core lands were created as an entity a bit before the year 1000 as a march (a border region of the Empire with certain special statuses) and given to Leopold I. of the Babenbergers by Emperor Otto II. in recognition of his help against a revolting Bavaria. In 1156 Emperor Frederick I Barbarossa issued the Privilegium Minus, which elevated the margravate to a duchy after some stuff with Bavaria again.

When the last Babenberger, Duke Frederick II, fell in 1246 at the Battle on the Leitha against the Hungarians it led to what is generally referred to as the Austrian Interregnum. Ottokar II Přemysl of Bohemia invaded in 1251 and got himself on the throne with local support and the hand of Frederick's sister. At pretty much exactly the same time we also have the general Interregnum period in the Holy Roman Empire after the dismissal of the Staufer Frederick II. as Emperor by the pope in 1245, which saw a period of internal power struggle for the throne until Count Rudolph of Habsburg managed to win over his main rival, Ottokar, in 1273 with the votes of all 7 prince-electors, becoming the first Habsburg Emperor... well, the 6 regular ones and, with the assumption that Bohemia would boycott, the vote of Bavaria instead.

Not one to let his Bohemian rival run wild, Rudolph claimed the Duchy of Austria for himself, eventually defeating the Bohemians at the Battle on the Marchfeld in 1278 and making his new Duchy of Austria, together with the other former Babenberg Duchy of Styria that had been taken by the Hungarians during the Austrian Interregnum, the new Habsburg core lands.

In 1356 Holy Roman Emperor Charles IV. of Bohemia issued his famous Golden Bull, which among other things fixed the 7 prince-electors elegible to vote for the Emperor (Trier, Cologne, Bohemia, the Palatinate, Saxony, Brandenburg and Mainz in that voting order) and increased their powers. Having extensive lands and a good power base in the Empire, the Habsburg were pretty annoyed to be left out and treated like a regular small duchy, so in 1358 Rudolf IV. had a forged document made that promised him rights similar to those of the prince-electors (primogeniture, their own customs and currency and immunity from the Emperor meddling with their courts, among other things). Because the forgers used the Privilegium Minus as basis and took its Imperial seal that forgery is now generally referred to as the Privilegium Maius.

Suffice to say with all the claims (including documents supposedly signed by Julius Caesar and Nero) that forgery was not acknowledged by the then-Emperor Charles. Fortunately, in 1440 with Frederick III. another Habsburg came back on the Imperial throne, who, in 1453, could just validate the Privilegium Maius himself.

So there is a lot of history behind the title, besides being an absolutely unique one. No reason to give that up. Besides, the Erzherzogtum Österreich ob und nid der Enns was still an internationally recognised and integral part and historical core of the Habsburg Empire.