I'll use the example of the Ancient Greeks to try and answer your question.

It's only relatively recently in human history that we've had an organized historical system, which is constantly evolving as new ideas and philosophies emerge. Our current, common understanding of history revolves around trying to piece together facts using documents, archaeology, and anthropology. We use written documents and things like tapestries, etc to draw conclusions about what we think historical facts are. Then those facts are compiled and we try to get a complete picture of a specific event or person by using those facts. That complete picture is what we think of as "history"

This was not always the case. Take Herodotus, the "father of history". Herodotus wrote in the 5th century BC and is the first person we know of to arrange writings and stories into a historiographic narrative. However, much of what Herodotus wrote has to be taken with a huge grain of salt. He exaggerates, embellishes, and sometimes outright makes things up in order to present a compelling and entertaining story because most of what he wrote were stories told to him second and third hand. As Herodotus was the first one that we know of to write history in this matter, it's very difficult for us to give a definitive answer to your question past that. A better place to ask this would be in the Archaeology or Anthropology subreddits, but the best answer I could give you would be: it is nearly certain that ancient societies pre-dating the Greek city-states (example: Mesopotamian cultures like those that founded Ur, Uruk, and Babylon) were very apt to pass down oral histories and oral stories from generation to generation.

Take "The Epic of Gilgamesh" - it is one of the earliest epics we know of, but by the time it had been written down, it had already existed as an oral story for many generations. That would be the primary method of passing down what we understand to be history during the majority of the "ancient" times. Being that they were orally passed down, we obviously have little to no records of this and can't definitively prove it one way or another, but did people in 15 AD know about the histories of people from 2015 BC? Yes and no - what they would have known past a certain point would have been cultural and local legends and little more than that (see: "The Aeneid", "Iliad", and "Odyssey", all of which were commonly accepted historical fact by Romans and Greeks but which we obviously know as being basic historical facts embellished/exaggerated by supernatural events and cultural legends)

China's a pretty interesting example here. Legend holds that the first emperors emerged about 4,000 years ago. While it's all but certain that some of these emperors are actually mythological figures, in recent years we've begun to uncover evidence that suggests the Xia dynasty might not be pure fantasy.

Now, it's entirely probable that even if the Xia dynasty did exist (the only thing that's been established so far is that some of the kings probably existed and that there were definitely early settlements where the Xia were supposed to be), what we know about them from the traditional narrative may be entirely wrong. But we're now learning more about what may have been one of the earliest proper polities in human history and that's kind of a big deal.

The interesting thing is that, if it turns out the initial studies pass muster (they're still being reviewed), they'll confirm, at least in very broad strokes, what we know about the Xia. Which means that historians and officials working around the time of the Han dynasty would, in fact, have known about the Xia from two millennia earlier. That's... kind of mind-blowing, when you think about it.

To be fair, the facts they believed they knew and what we'll be able to verify will differ rather dramatically. But that's always the case when you're talking about very long stretches of time. How much did we learn from the ruins of Pompeii? And those excavations didn't really begin in earnest until the 18th century. We wouldn't have been able to learn the things we did if we hadn't forgotten that knowledge in the first place.

Unfortunately, most cultures didn't really develop the kind of record-keeping that would be ideal for answering this kind of question until much later than c. 2000 BC. Not all cultures valued writing things down like we do. Oral traditions have always been popular among migratory people. We're probably not going to find a 4,000 year old Evenk library buried under the frosts of Siberia. No matter how cool that would be.

Isn't it possible, that, in say, 2,000 years, people will not know much about us because methods for storing information about the past might become quite more advanced? We know that if those 2,000 years ago had internet and other modern technology, we would probably know a whole lot more about them. But, in the year 4,000 (2,000 years from now), I wonder if the way we keep track will actual tell very little about us, compared to say, those who live in the year 3,800, only 200 years before reference year in question.