Could you have similar companies appearing during the American colonization?
The problem here is that the imperial, Austrian branch of the Habsburgs never owned ports near the North Sea. Triest on the Adriatic Sea was made a free port in 1719, the same era that saw the creation of the KOK. The Triestine East India Company, founded by Maria Theresia, existed from 1775 until 1785. They acquired a port at the Delagoa Bay in Mocambique and the Nicobar Islands for a short time.
Basically, the Austrians respected the Spanish rights until the War of the Spanish Succession. Given that even Flanders and Aragon had no rights to trade in Castilian overseas territories, it is hardly imaginable that the distant cousins in Asutria will be privileged.