Why wouldn't they?

Things the DPRK has: a parliament, representatives, an electoral system linked to regional and municipal government. Now, the independence of these structures is provably nonexistent, and real political power is entirely concentrated in the hands of the top slice of songbun, the core class, but why should that have any impact on a country's self-definition as a democracy? We don't begrudge a fledgling United States the term democracy, despite the fact that political power was reserved exclusively for white, landowning males. More to the point, we don't seem to take issue with the Republic of Korea having claimed to be such through its almost four decades of military dictatorship.

Historically speaking, the foundation of the DPRK and the ROK is important because it represents a long sought-after goal of Korean independence. Remember that the for Korea, the 19th century represented the growing colonial interests of the three major powers of the region: China, Russia and Japan. After a number of conflicts left the country in Japan’s sphere of influence, Korea was formally annexed by Japan in 1910. What followed was the colonial period, an often brutal era in Korean history in which the nation and its people were co-opted into Imperial Japan’s expansionist designs in the Pacific, with much of their land and industry taken over by the colonial regime. Even the nature of the Korean identity was under assault during this period, with Koreans expected to take up Japanese names, to teach the language to their children.

So the end of the Japanese occupation was, quite literally, a return of power from colonial forces (the Japanese) to Korean ones, even if on both halves of the peninsula these forces were propped up by foreign interests. It should be unsurprising that governments arising from this juncture in history would stress their democratic and republican qualities: finally, Korea would be governed by its own people again.

As for more specifics on the names of Korea, while I've never come across sources discussing why 'Chosŏn Minjujuŭi Inmin Konghwaguk' was specifically selected over another name without the word 'democracy' (minjujuŭi), it should be noted that a primary agitator for liberty from Japan in the colonial era was the Provisional Government of the Republic of Korea, a government in exile based in China.

This name is not quite a direct translation, unlike the DPRK's: in Korean, it is Daehan Minguk Imsi Jeongbu, meaning 'Great Han People's Nation'. While Korean culture and history is often intricately interwoven with that of the Chinese, don't mistake this for a reference to the Chinese Han people or dynasty, it's instead related to three Korean confederacies (Samhan) that date back to the first centuries of the Common Era.

The provisional government was responsible for the coordination of a lot of disparate Korean partisans, but after the collapse of the Japanese colonial empire in the wake of WWII and into the short period of trusteeship to follow, those bonds were quickly broken. The Provisional Government (with future ROK president Syngman Rhee as its leader) took up residence in Seoul, and became Daehan Minguk, the Republic of Korea. On the northern side, the Provisional People’s Committee was established by the Soviets with none other than Kim Il-Sung at the helm.

It makes sense, then, that a name was chosen that would differentiate the northern half from the southern one: where the south was the Republic of Daehan, or ‘Koryo’ (where we get the word Korea), the north was the Choson Democratic People’s Republic. This is as much as I’m aware of with regards to the name of North Korea.

The primary source for this post was Michael J. Seth’s A History of Korea: From Antiquity to the Present. Please note that if you’re interested in picking up this book, avoid the abridged edition that stops at the modern era, and thus will not include the majority of the above information.