Since 'Chinese' is an impossibly complex and ever evolving category, and the act of defining it always leads to problems, (from being noncomprehensive, to being circular, to being racist etc.) it is true that every regime that ruled in the land that is China seemed to add some component to that ever changing idea. While I think it is best to dissolve this question, the act of answering it has some merit and if it interests you, or shows you a more complex picture that leads you down a path of interest in Chinese history, then I shall attempt it.

To get it out of the way, I'm not confident why Chinese people refer to the Han dynasty, but I will say it is not so clear cut. There are still Chinese diaspora communities that refer to themselves as Tang people, and in some Sinitic languages, they refer to themselves as Tang. You further asked about what cultural distinctives that Han society mixed with, but this is actually a very hard question to answer and is so broad you might consider asking this in a separate thread.

As to the layers, I will try to give a broad sweep, though specialists in other portions of Chinese history might have different focuses or disagreements. (I myself know only a bit about the Zhou). Most of this will be taken from textbooks used in survey courses on Chinese history, and so will be very broad and general, but if you have more interest it may point you where to look. Roughly,

1. The Shang are the first regime/dynasty that we can identify seemingly Chinese characteristics in the historical record, such as ancestral worship and divination. There are other kingdoms and neolithic peoples that came before that we might also find some common elements as well, but the Shang is commonly cited as the first one, and its one we have evidence for, though there were contemporary cultures such as the Ba and Shu people and the excavations from Sanxingdui is enriching our understanding of non-Shang people's contribution to Chinese culture.
2. The Zhou established the Mandate of Heaven, broad kinship networks, and the first literary works we know of. Bureaucracy also seemingly begins in this time. Warring States saw the establishment of sovereign territories and the precursors to imperial control as well as the birth of Chinese philosophy including Confucianism, Daoism, Legalism, as well as others such as Mohism. We also see elite common culture and a Huaxia culture being defined in relation to other people such as the Rong and Di. It is also common for many to peg the Zhou as the beginnings of something distinctly Chinese
3. Qin and Han empires gave us institutionalized Confucianism and foundations of imperial governance. Much of the territory they take, while today are Chinese, are not by their standards, civilized. We see the enshrinement of a literary culture and preeminence that they use to effectively define themselves as civilized. A person, no matter from where, should they have the same pool of knowledge and literary understanding, would be civilized in their view. A gentry emerges here.
4. Cao Wei through the Northern and Southern dynasties: we see the emergence of the south as a significant region of China, and the literary and poetic forms reaching what is considered their height at this time. Confucianism even fades for a time, with Daoism and soon after Buddhism taking its mantle.
5. In Sui and Tang we see a very cosmopolitan empire, one that has a variety of peoples involved. Landlordism and aristocracy have made their return here. We also see the opening of the civil exams to everybody, though not all have the opportunity to take it. Broad trade networks are established here, and Buddhism is at its height. The 'tributary system' also begins.
6. In Song and the 'Conquest' Dynasties: we see a challenge to what even is Chinese. The Song are relatively meritocratic and prioritize the civil over the military, all while other powers are able to claim the Mandate of Heaven and say they are the pinnacle of 'civilization'. Neo-Confucianism has its roots here, unifying Confucian and Buddhist thought. The south perhaps reaches its highest prominence here.
7. The Yuan and Ming encourage the growth of Neo-Confucianism and things such as footbinding become the norm. Mandarin, now having split from the Sinitic languages is prioritized and we see the full prioritization of the landed gentry. We have much of the great works of Chinese novels introduced here, and the expansion of alternative forms of media such as opera as opposed to just literature.
8. The Qing is China at the height of autocracy, and many of the territories that are in modern China are first conquered here. From here, we see the growth of Chinese nationalism and the introduction of western originated things, such as Christianity, Science, Philosophy, etc. Lots of Chinese Diaspora communities arise from here on out, and with the Qing ruling, our definition on what is Chinese can only become more and more complex.
9. Beyond this, it's a little hard to say what exactly are the effects of modern regimes on Chinese culture. It becomes controversial as well, with Taiwan and mainland China's situation. Then there are the diaspora communities, and globalization, and China's geopolitical projects.

And even in my quick summaries, I feel guilty leaving out a lot of things that other historians may feel are equally if not more important to Chinese culture, such as women's rights, cooking, fashion, military, queer and sexual history, economics, agriculture, science, historiography, colonialism, etc. There's just an infinite onion's worth of layers to peel back here, and there's no core.

The question itself is ill posed. At what point can we say something is Chinese, or what is more Chinese than not? Even the identity of Chinese didn't really come into being until the nationalism that began during the Qing. The answer to this question is very fascinating, but the question itself has no definitive answer, and might be best left unanswered.

Sources:

I will just recommend survey textbooks, or good combinations of books to get your fill of Chinese history and these claims

1. Valerie Hansen's The Open Empire and Jonathan Spence's The Search for Modern China . This is a good combo, with Valerie Hansen being a good sweep from Ancient China to the Ming dynasty. Jonathan Spence's book covers the rest. If you want the quickest introduction to Chinese history in general, these two are the way to go.
2. Harvard Press's History of Imperial China, edited by Timothy Brook. This is a very up to date 6 book introduction to China through the dynasties, but doesn't really cover the Shang and is more focused on the imperial era from Qin to Qing.
3. For the most hardcore, Cambridge History of Ancient China and Cambridge History of China is the most comprehensive English resource on all things China. Unfortunately, volume 4 remains unpublished, but the rest of it is all there, and while some parts are now out of date. If your interest is strong, this will be a fun read
4. Beyond Huaxia podcast by Justin M. Jacobs is a more casual overview of China.