Balkanize as a verb just means to break up into several smaller states, often but not always based off common culture/language/religion, similar to what happened to both Austria-Hungary, the Ottoman Empire, and Yugoslavia.
It was first used after the first world war to describe what had happened to the region post Austria-Hungary and Ottomans/the balkan wars that preceded ww1. Darkly humorous that the balkans themselves have been balkanized more than once, though.