Sorry I don’t have my sources on hand right now. Sources are in Chinese. Will edit later to add.

The designation of "Han" originated from people belong to the Han country during the Han dynasty. The Han people and culture originated from the middle plains (中原) of China. Many names have been applied to this group, depending on the name of the government/ dynasty at the time.

Modern day “Han” includes many groups of people who are not descendants of the Han from the middle plains. Generally, the movement of culture of Han culture in history is form north to south. This is evidenced every dynasty being formed by a group from the north, conquering south-ward, with the exception of the Ming dynasty (明).

There are several groups assimilated into the Han culture in history. The most well-known example is probably the Xianbei (鲜卑). They were a northern ethnic group in China most prominently known during the 16 Kingdoms period (五胡十六国) and the north and south dynasties (南北國). 魏孝文帝 was the king of北魏, a Xianbei nation. During his rule, he pushed for policies that promoted switch to Han culture and practices, including changing his Xianbei family name to a more Han sounding name. These policies reduced cultural clashes between the Xianbei and Han, also stimulated trade with the southern Han nations. Furthermore, the Sui and Tang dynasties, which followed the north and south dynasties, are considered as Han dynasties today, even though their founders were nor of pure Han descent, because of the these policies that adapted the Han culture.

Another example is the Haklo people, who are the majority in present-day Minan and Taiwan. Haklo are considered a branch of Han today, but originally came from the Yue people (越族) who had adapted the Han culture.

There had been studies that found the northern Han and southern Han to form distinct genetic clusters from each other, suggesting distinct genetic origins. The Han culture is like a wide umbrella term compassing many branches and regional variations. However, these various branches share similar lores, origin stories and myths in broad strokes, suggesting a common cultural lineage.

Personally, I think the idea of defining a cultural singularity arose from Western nationalism, and this makes the designation of “Han” a constructed one. After all, why “Han”? Why not “Ming,” which is the last dynasty governed by this cultural group? However, the culture itself is not constructed.

Now to address the comment from /u/bitparity: It is a very interesting phenomenon that Han people rarely realize that they are Han. In fact, there are grass-root movements for Han people to re-discover Han culture and attire in China right now. Of the 50-something official recognized ethnic groups, Han is the only ethnic group that did not have an officially recognized traditional attire that reflects its culture in China, demonstrating the lack of connection and understanding of modern Han people to the Han identity.

My speculation is that the Han is so predominant in China, that it is the de facto “norm,” and thus does not require a designated identity. I liken this to how the deaf community refer to the rest of the population as “hearing,” but few non-deaf people would refer to themselves as “hearing.” The same goes for seeing people. Since hearing and seeing make up the majority, they are the norm. With the way society works, names are given to minority groups, but rarely to the majority group. I think Han people don’t call themselves Han, because they don’t realize there’s a name for what they are, not because Han is a constructed culture.

Is the Han identity in Korea the "same" Han as China?

An introductory study on the ethnogenesis and evolution of Han identity can be found in Mark Elliott's "Hushuo 胡說: The Northern Other and Han Ethnogenesis". The short version is that an ethnic identity known as 'Han' has existed since at least the early 6th century, when it referred to the natives of Northern China in contrast to the Xianbei and Xiongnu migrants and invaders, but that significant changes occurred to the definition and scope of this identity over the next 1,500 years. The modern definition of Han identity mainly developed during the Ming and Qing dynasties, when again it was used to contrast the natives of China - now both northern and southern - with groups like the Mongols, Muslims, and Manchus. It was then formalized by the ROC and the PRC as an official ethnic identity in the Western sense.

For more detailed studies on the development of Han identity during various historical periods, there are many scholarly options, including Mark Edward Lewis's China's Cosmopolitan Empire, Abramson's Ethnic Identity in Tang China, Patricia Ebrey's Surnames and Han Chinese Identity, etc. I would, however, highly recommend checking out Yang Shao-yun's thesis, "Reinventing the Barbarian: Rhetorical and Philosophical Uses of the Yi-Di in Mid-Imperial China, 600-1300" as it is probably the most comprehensive discussion in English of the history of Chinese identity, "Han," "Hua," etc. for the period mentioned and more besides, and represents the most up to date understanding of Chinese identity politics. Be prepared, however, for a long and technical read.

Long story short - all ethnic identities are constructed, mostly through the process of state building and Us vs. Them interactions, and the idea of a "natural ethnic identity" is an oxymoron. The Han ethnic identity might have a longer history than many others, but it would be foolish to assume that, unlike other ancient identity concepts like Persian, Indian, Greek, etc., it has not undergone significant changes since its inception.