In Redcoat: The British Soldier in the Age of Horse and Musket the historian Richard Holmes writes on p. 103:
Most European armies maintained bodies of Household troops - Austria was a notable exception - in which birth and breeding were prized.
So why not Austria? They certainly set as much store by birth and breeding as anyone else in that era so it cannot have been on any democratic principle. I suspect some crazy quirk due to the Habsburg monarchy being even more patchwork than common but cannot even begin to form a specific guess.
There were five guard units formed by Maria Theresia in 1763, which were considered household troops in Austria-Hungary. These are the:
- königlich-ungarische Leibgarde
- k.u.k. Trabantenleibgarde
- k.u.k. Leibgardereitereskadron
- K.u.k. Leibgardeinfanteriekompanie
The first two followed the Emperor around and were in place for representative purposes. For the Arcièren-Leibgarde, especially decorated, tall enough, good-looking enough, until 1780 Catholic and until 1807 only royal older Officers were chosen. However, the first two guard regiments were not big enough to perform any significant military operations.
The latter three regiments were likewise smaller groups (a maximum of a few hundred), but also consisted of lower-ranking military mebers. The Trabantenleibgarde performed guard-service at and around the imperial residence in Vienna (Hofburg). The Leibgardereitereskadron performed ordonnance-services, mostly as couriers. The Leibgardeinfanteriekompanie was tasked to keep order around the imperial residences and gardens and the castles Schönbrunn and Laxenburg.
At least regarding the first two guard regiments, optics were everything, which would have included "birth and breeding", although their size might have been smaller than for equivalent other powers, which may be what Richard Holmes is alluding to.
A background to its comparably small size and late adoption is likely (aside from Austria-Hungarys multinational character) because Maria Theresias military reforms attempted to modernize the mainly imperial (i.e. Holy Roman Empire) army into a more Austrian army and reduce inefficiencies, among them over-reliance on the feudal nobility. Wilhelm von Haugwitz tasked with this change tried to change the army away from a feudal aristocractic instutition to a more modern army, which would influence the size of the created guard units.
Sources on German Wikipedia: https://de.wikipedia.org/wiki/Arci%C3%A8ren-Leibgarde
Sources on Maria Theresia's reforms: https://en.wikipedia.org/wiki/Maria_Theresa#Reforms
Sources on Haugwitz: https://en.wikipedia.org/wiki/Friedrich_Wilhelm_von_Haugwitz