Let's answer your second question first, shall we? It's tied in with the answer for your first question, so why not start there?

did people care as much about history back then as they do in this era?

Yes. Oh god yes. Faaaaaaaaar more. But for totally different reasons.

In the modern world we view history as the study of written (and, increasingly) archaeological evidence of past events and cultures, in order to discover basic truths about the development of humanity and the modern world, as well as basically just figuring out what makes people tick. Ok, well that's interesting, but not fundamentally important. I mean, the world won't end (probably) if you as an individual don't study history.

In the ancient world, the world would end. "History" for us is primarily based in written records or quantitative archaeological evidence. It's something that we know happened because there was a record of it. That's not really the way it worked in antiquity, at least not until after the Ionian historiographers and Herodotus. In antiquity we should really ask whether knowledge and study of the past, both as it was actually recorded and as tradition passed it down, was as important. And it was. Way more. Because ancient societies are built on orders of tradition. There are traditional behaviors, traditional rituals, traditional codes of social interaction. And those traditions are grounded in some form of origin story, which, whether actually factual or not, was assumed to have actually taken place. Understanding of a society's origins was of paramount importance, and certain societies were obsessed with understanding of their own origins (the Romans, for example, had this unhealthy obsession with figuring out exactly what it meant to be Roman). Understanding of one's own origins was crucial to the maintenance of social rituals that kept the culture cemented. So in a very real sense, without a clear understanding of the basic "history" of a society, communities would quite literally fall apart through loss of tradition.

Now, that sort of answers your first question as well. Societies regarded legendary and semi-legendary events, generally taking place before any system of writing was able to codify and confirm events, as being their own "antiquity." There are certain examples, the most common of which that is cited on the sub is usually Hesiod's story of the Ages of Man. But beyond something like that, any sort of local origin story was taken as fact, at least for the sake of ritual. This could be a local origin story, like the story of Romulus and Remus (the origin cult of the Latins) or the story behind the cult of Aeneas (a cult derived from the Etruscans). It could be an ancestral story of a single family (there are lots of these, but one good example might be the story of the Alcmaeonids, who claimed descent from a descendant of Nestor). Or they could be the all-important universal "histories," such as the Trojan Cycle, that cemented the ties between related localities and joined them into a single larger social group.

But there were other histories as well. Following the development of what we'd call "true history" there was a boom, particularly during the Hellenistic Period, the 1st Century, B.C., and later in the 2nd Century, A.D. of professional histories. The ancients thought of written histories as being always universal, starting from earliest times, unlike our more common histories of short-lived events. So we see the Hellenistic historians compiling enormous piles of origin myths from different peoples and listing them out one by one before starting on the next round of subjects. Even Herodotus does this, by going through the early histories of each of the major players in his narrative before embarking at the very end on the Persian Wars, which are the crux of his work. Hell, even Thucydides (who was often not, along with his imitator Sallust, or even Xenophon, considered to be a true historian, but something more like a chronicler, since he didn't write universal histories but focused on specific events) feels compelled to give a very brief rundown on the history of political strife among the Greek states, although he does this mainly to explain the causes behind the monumental disaster (i.e. all-out shitfight) that was the Peloponnesian War. Ancient historical works are strange for us, not only because of their use of rhetoric and thematic material, which is stressed over precise factual accuracy (partly because it's awfully hard to be factually accurate about things that happened hundreds of years ago when people didn't make detailed records but simply said one army beat another army end of story), but also because despite the fact that they are "universal" histories, they are still in their most crucial material contemporary histories. Ancient histories relied almost entirely on verbal evidence, since the detailed bureaucratic records that we are used to today, although in use, were used to record other things. So they are universal in scope, but contemporary in content. And in any case there are "chroniclers" like Xenophon, Thucydides, and Sallust, writing on contemporary events that they had witnessed and that were very narrow in scope (basically cherry picking the endings of most universal histories and cutting out what they would've believed to be of little importance), who are often considered of greater historical value to modern scholars, although often of inferior literary value (although Thucydides is a marvelous writer and was really a genius. Yes, I kind of have a hard-on for that guy.).

So what would the average person have thought about, say, Herodotus? Well, he'd be intensely interested in remembering and understanding events from his own lifetime, which was honestly the important bit. Histories were often said to be written for posterity lest the memory of the events disappeared, but it wasn't really true. Historical works almost always, in the onus of their content (the contemporary material) were attempting to communicate with the contemporary readers who had experienced the events in question, to try to interpret what had happened and why it was important. Their value for posterity did not diminish really, but that wasn't actually the intended audience. As for Herodotus' discussion of, say, the history of Egypt, would an average person have cared? Probably not. Books were expensive, at least keeping them was, although undoubtedly many people frequented the great public libraries to get good access to many works. So much of that would've been of sort of eccentric interest, as an interesting oddity much like the early museums of the Enlightenment, entertainment and brain fodder for an educated elite (and, at least in Herodotus' case, much of that material was included for that purpose, as an interesting oddity. Historians often considered themselves first and foremost storytellers. They told stories that, to the best of their ability, they believed were more or less true, but they still were telling stories). Still, many people would be aware of the content, particularly the stories that pertained to their people, but they wouldn't have been interested really in figuring out exactly what happened long ago in some far away place.