To understand the language situation in Korea, let's imagine that we divided the Italian peninsula in half just north of Rome. Does it follow that now there are two dialects, North Italian and South Italian? No. Italian has a bewildering variety of regional dialects, and our arbitrary line doesn't correspond to a genuine dialect border. Similarly, the DMZ cuts across the central dialect region in Korea.
More importantly, Italians will continue to write and be taught in Standard Italian, which was developed based on the Tuscan dialect long before our artificial division of the peninsula. It won't be as if they would start from scratch and create new standard languages based on the Milanese dialect in the North and the Roman dialect in the South. Even independent countries such as Germany, Austria and Switzerland find it useful to use the same standard German as each other, even if it's not necessarily based on a dialect spoken wothin their borders. There will inevitably be differences in vocabulary and spelling, but the differences will be far less than if we imagined a naïve model where each country creates its own standard (which is what basically happened in Scandinavia).
More importantly, Italians will continue to write and be taught in Standard Italian, which was developed based on the Tuscan dialect long before our artificial division of the peninsula. It won't be as if they would start from scratch and create new standard languages based on the Milanese dialect in the North and the Roman dialect in the South. Even independent countries such as Germany, Austria and Switzerland find it useful to use the same standard German as each other, even if it's not necessarily based on a dialect spoken wothin their borders. There will inevitably be differences in vocabulary and spelling, but the differences will be far less than if we imagined a naïve model where each country creates its own standard (which is what basically happened in Scandinavia).