Primogeniture is common, but it's not strictly followed either in other parts of the world. For example in Saudi Arabia they follow agnatic seniority where you pass to the current ruler's younger brother instead of son. In ancient China primogeniture is sometimes followed, or the ruling emperor might select an heir, for example Emperor Kangxi of China has the name of the heir in a sealed box only to be announced upon his death to avoid conflicts. (Both system works better when the ruler has many wives producing many sons). An example from Korea would be Sejong the Great, who is the third son. He became heir apparent after his brother was removed of his position of heir apparent.

Although in modern times we don't actually know if North Korea has a strict rule of succession. After all Kim Jong-un has a middle older brother Kim Jong-chul who got passed over as well.

Was primogeniture observed in Korea historically or is it more of a Western precept?

Yes but not always. In the Goryeo dynasty, lineage was traced through both male and female lines. By the time Taejo took over and established the Joseon dynasty, Korea adhered to a strictly patrilineal descent system after years of dedicated Confucianization. The Five Classics and Four Books popularized the concept of primogeniture in the Sinosphere. But the Ming dynasty, which ran concurrent to the Joseon dynasty, had relaxed the custom to the point of being virtually ignored. Inheritance was split equally between all sons regardless of birth order, with the eldest receiving slightly more as a nod to tradition.

In contrast, Korean society singled out the eldest son as the exemplar of his entire generation, resulting in a specialized focus on the main line at the expense of cadet branches. Equal inheritance still happened in Korea but it no longer became the norm as it was in the previous dynasty.

It's difficult to gain any information on how much Confucianism is integrated in North Korea but one of my close friends is a North Korean refugee. From his descriptions of the country, I don't think Confucianism plays a significant role in North Korean culture.

Source:

The Confucian Transformation of Korea: A Study of Society and Ideology by Martina Deuchler

Not entirely sure about Korea historically but it may have followed the general Sinosphere rule.

In China primogeniture during Imperial times was generally followed and endorsed by Confucianists, but it was more a traditional/best practices rather than a law there were so many exceptions, including some of China's historically most exemplar and well regarded emperors, that the tradition was frequently challenged.

There were three general ways to name your successor, in order of prevalence:

The proper son. Meaning the son of your current formal wife, as opposed to a concubine.

The eldest son (which can often not be from the formal wife!).

The most talented son.

All three traditions are fairly well accepted as reasonable and vassals/courtiers would generally have no problem accepting either of these three choices by the ruler (of course the "talent" thing is very subjective. It's generally the best scholar and politician in the family who is viewed as the most talented) Of course if your son by your formal wife happens to be the oldest amongst the lot and also the most talented then that's jackpot.

source: Yi Zhongtian, Pin Sanguo (Savouring the Three Kingdoms)

Keep in mind, North Korea is technically a Communist state. Stalin's son didn't become leader after Stalin's death. The current leadership of China are not the children of the former leadership. Inheritance, however, is how things practically work in North Korea.

Propaganda maintained that Kim Jong Il was the best person for the job - that it was actually harder for him to become leader, because people would think it was only because he was the son of Kim Il Sung, not because of his amazing personal qualities and because he was the most qualified. I'm not sure how propaganda managed this latest transition. The fact is, however, that those in power have an interest in keeping this system in place. There is a joke in/about North Korea -

"Can the son of a Colonel become a General? No, because Generals have sons too !"

I am reading Genghis Khan and the Making of the Modern World by Jack Weatherford and it seems like, at least in the Steppe tribes tradition, primogeniture was the norm. Repeatedly it states that the eldest son, once the father dies, is expected to take place as the head of the family. In fact, Temujin (who would eventually become Genghis) killed his older brother (from a different mother) with the help of his smaller brother since, when their father died, he was going to take his place, and even claim Temujin's mother as a wife. So at least it's not a purely western concept. Mongolia and Korea are fairly close to each other, although their cultures differ.

Later on, when Genghis passed, the book states that even though the Mongols held elections for the office of Great Khan, at the summit where they met to choose their new leader, eldest sons had the right to make the first toast, boast, and speech in general... although they were usually challenged on that right. But that is another matter.