People have had more or less the same base texts since the Renaissance. The renaissance itself was a process of textual gatherings, so you shouldn't imagine that as soon as 1400 rolled around everyone had all the texts we do today, particularly Greek texts, but the antiquarian activities of the Renaissance humanists created the basic body of "classical" texts we have today.

Now this gets more complicated when you look farther back, because the humanists did not actually "discover" texts, or rather they only did so in the way Columbus "discovered" America, because obviously every text discovered by the humanists had been passed down through monastic libraries. So Livy, Cicero, Suetonius etc were all floating around in the Middle Ages, but in general knowledge was more diffuse and history in general was not as interesting as philosophy. So while clearly somebody was enjoying Livy, it wasn't really a text everybody read.

That being said, this does not mean the historical events and personages were not of interest, they were simply adapted to suit Medieval forms, and I would say that virtually all of the anecdotes that are somewhat part of "pop culture" today would be known to the educated, and even beyond in the form of ballads and songs. Fir example, versions of the highly adaptable Alexander Romance were wildly popular, and collections of stories such as The Golden legend contains stories of exemplary figures from Antiquity such as Trajan. Chaucer is a great example of this, containing numerous references to classical anecdotes.

While I would shy away from calling the widespread knowledge of Caesar's assassination or Socrates "history" as we think of them today, anecdotes were widely passed around and popular. Which really isn;t so far from what it is today.

Caesar: A Life in Western Culture by Maria Wyke gives a very focused discussion on this in regards to the well known anecdotes of Caesar's life.

To add to Tiako, some ancient writings were more popular than others, and those may have a larger effect on the historical events and mythical stories we allude to than any other reason.

For example, there are over 200 copies of Orosius's history from the Middle Ages. He wrote a 7-part history from creation to his present (early fifth-century). In comparison, there is only one copy of Tacitus's Annals (the premier source for the Julio-Claudian Era). This was a didactic attempt to show how Christianity made everything better even though it seemed to many Romans that the empire was crumbling just after Christianity became the state religion and post hoc, propter hoc. Most modern historians wouldn't use this as a source for anything before Orosius's own time, and even the part that deals with his own time is incredibly biased and he fabricates narratives to fit his argument, so it's generally considered "bad history".

However the Middle Ages loved him. He had a biblical and religious slant, but more importantly he was able to fit the history of the world into 7 books (each being maybe 50 pages in a modern printing). This meant that they could appreciate their history without using tons of parchment and ink. To cover the same scope, first they would need to learn Greek since Romans weren't particularly interested in earlier history, then they would need to combine the Bible, Livy, Josephus, Ammianus Marcellinus, Tacitus, etc. which would amount to hundreds of books (Livy alone had 150 books covering Roman history beginning at 753 BC). So they settled on short, sweet and readable Orosius.

Because of his popularity, our narrative of ancient history is still colored by Orosius. We jump from the Near East to Greece to Rome in chronological order just like he does! When you took/teach a survey course in ancient history, did your textbook mention anything on the Persian steppe after Alexander? Or anything in Egypt between Ramses and Ptolemy? Or anything in Greece after Phillip II? In large part that's due to the narrative style Orosius made popular. History was split between protagonists as a way to simplify it. (You can contrast this method with Diodorus or Polybius who hop around the Mediterranean giving the events in the Greek world and the Roman year by year instead of focusing on one or the other).

So what does this have to do with your question? Well our view of antiquity isn't the same as the way the ancients viewed themselves in large part due to the economy of the copyists. While they could reference Virgil with the best of the Romans, many of Cicero's allusions (he died before Virgil flourished) are barely understandable now because he loves to reference Roman historical dramas that are entirely lost because the art form had fallen out of favor (probably long before the monks).

So there is a lot of material that has the potential to really change our view of Roman history. Knowing what their plays were like would give us an insight into political discourse. Great discoveries have happened that lead to entirely new understandings of ancient culture. The Derveni papyrus discovered held in the hands of a corpse in 1962 has totally transformed our knowledge of mystery cults, for example.

Probably most influential discovery was Tacitus's Germania, discovered in the fifteenth century. It created the idea of a German nation (there was no "Germany" to speak of at the time) that would inspire Nationalist movements in the nineteenth century and racial purification in the twentieth. You could say that the rediscovery of Tacitus created the meme of nationalism. For more on that check out A Most Dangerous Book by Christopher Krebs.