Via Snarkmarket, I found this (probably incomplete) Wikipedia list of the oldest companies in the world that are still operating today under the same name. The oldest one, a construction company in Japan called KongÅ Gumi, just went belly-up after serving their customers since the year 578AD.
And according to a commenter there, the oldest University in continuous operation is University of Al Karaouine in Fes, Morocco.
The oldest company on the list from the Balkans is Apatinska Pivara which has been brewing beer continuously since 1756. They produce one of the most popular local beers, the Jelen Pivo (although, both in the region and for emigrants like me, the champion is Montenegro's Niksicko Pivo, both the pilsner and the stout).
WWII and the subsequent nationalization of all sorts of businesses makes it unlikely that many old companies continued under the same name afterwards, but I cannot believe that only one beer brewery made the list. Anyone here from the Balkans can think of (and verify) some other companies with a long tradition?
A lot of breweries wouldn't qualify because their claim to ancient heritage is based on there being a brewery there or thereabouts, not that the company itself has existed continuously all that time.
For example, the makers of Stella Artois claim to have been around since 1366, but in fact all this means is that beer has been brewed in Leuven since then. My best offer would be Shepherd Neame in Kent, which has been going since 1698.