This answer honestly is reductive because I cannot lay out a book length response that would do this justice, but the best way for you to think about how the study of the ancient world came about is to first conceptualize orientalism and colonialist mindsets. The academic discipline of ancient history was primarily a result of colonialist expansion and fantasizing about the "Orient" that is, the "East" as though it were this mystical place of wonder and magic, hidden treasure. Now, of course, the reason why Chinese ancient history is not called "ancient history" over in our studies, is because Colonialist history was primarily centered on: how did we white men come to where we are, which primarily centered on the promotion and intellectual worship of Greco-Roman and Germanic cultures specifically, which formed our "western world", while the rest, by and large, was shifted to... "the orient." Now the Middle-East got included, however, because Christian religion stemmed there, and therefore we had to find our "ancient roots" there as well. Hence, "archaeologists" went around, as is commonly said, with "a Bible in one hand and a spade in the other." A great read on the notion of orientalism and specifically how it has affected people is Edward Said's, Orientalism (Pantheon, 1978). While old, it is basically considered a classic in postcolonial studies. However, this is also similarly why we pay very little attention to the ancient history of Slavic peoples as well, and thus, there is very little written on (for example) pre-Christian Slavic religion in English languages. In fact, the majority of what I can find is written in Polish and Russian. Slavic people were not considered traditionally "white" or properly "European" like those of the (to echo General Robert E. Lee of the Confederacy) "Anglo-Saxon race".

Thus, there is a division of East and West that develops, and both of them are largely mythical constructs. The reality is that there is no "Western civilization", this is just a mythic concept created by white colonialists, which has since just caught on. And thus, people of the "Eastern civilizations" were often seen as sub-human, barbarous, savage, etc. Essentially, the same way we ended up envisioning Native American civilizations. And thus, they were not "ancient" in the same way that we were.

So, the reason why we don't traditionally think of Chinese history when we say "ancient history" here in Euro-American nations, is because of this. Colonialist and Orientalist mindsets, steeped in generations of racism and white supremacy, along with the fact that Colonialist theft of artifacts, fantasizing and eroticizing the "Orient", and more has been exceptionally profitable for many pockets (as an example and controversy, look up the Museum of the Bible, Dirk Obbink, and other similar controversies recently).

For other books I would suggest:

Douglas McGetchin, Indology, Indomania, and Orientalism: Ancient India's Rebirth in Modern Germany (Fairleigh Dickinson University Press, 2009)

Kiri Paramore, Religion and Orientalism in Asian Studies (Bloomsbury, 2016)

Steven Holloway, Orientalism, Assyriology and the Bible (Sheffield Phoenix, 2006)

Bonnie Effros and Guolong Lai, Unmasking Ideology in Imperial and Colonial Archaeology: Vocabulary, Symbols, and Legacy (ISD Uni. of Cali., 2018)

Daniel Varisco, Reading Orientalism: Said and the Unsaid (University of Washington Press, 2017)

I deleted a previous answer I was writing, but I would seriously question the premise in the OP.

I won't deny that historically for European and North American historians "Ancient History" or the "Ancient World" very specifically was the Mediterranean and Near East, but saying that museums and history books in the current day as a whole still follow this paradigm sounds misleading.

I checked the British Museum, Louvre, and Metropolitan Museum of Art, and they don't classify their collections this way, although they do talk about the Ancient Near East, or Ancient Egypt (Greek and Roman collections are separate from these, by the way). The Boston Museum of Fine Arts did stand out with an "Art of the Ancient World" exhibit, which focused on the Med, Near East, Egypt and Nubia (which is interesting because MFA has quite a bit of ancient Chinese and Indian art as well).

Barnes and Noble's "General Ancient History" category does have a big focus on the Near East, Greece, Rome and Egypt, but not exclusively. General books like The History of the Ancient World: From the Earliest Accounts to the Fall of Rome by Susan Wise Bauer most certainly include sections on China and India, or The Penguin Historical Atlas of Ancient Civilizations, which covers the Near East, Mediterranean and Egypt, but also East Asia and even the ancient Americas.

Looking through results for "Ancient History textbooks" online, I do see a few "Ancient World" books that still go by the Med+Near East paradigm, but they're outnumbered by textbooks on ancient civilizations that include East and South Asia, as well as ones also including the Americas and Africa (and treat ancient history as a subsection of world history, rather than regional history). I tried looking up my ancient history textbooks from...longer ago than I care to admit, and I can't find those in particular, but they were also written in the latter vein, so treating Ancient History as a period of World History in general US education at least would be several decades old at this point.

Which is all to say, I don't want to disagree that there was a time when "Ancient History" specifically referred to events and civilizations in Europe, Egypt, and the Near East/West Asia, but I question that this is still some sort of universal paradigm in education, historic writing, or museum collections, even if it occasionally remains a shorthand for that period in those regions.

ETA - part of my reaction is because I feel like in the past few decades there has been a lot of writing on the Ancient World that, for lack of a better term, could be labeled as "Silk Road histories", with a focus on the economic, cultural and political connections across Eurasia in the ancient period. Peter Frankopan's The Silk Roads: A New History of the World is a popular example of this, although that type of history isn't quite as "new" as Frankopan would have you believe. There have been a couple decades' worth of comparative histories of Rome and Han China, for example.