Aside from colonies which obviously will have a different culture by virtue of being a colony, and they're also newer compared to their host countries, were there small regions that functioned as particular areas of a certain culture? Both the Roman Empire and the Egyptian dynasties lasted for a really long time so surely something arose.
Unless the education regarding other parts of the country/empire wasn't very good for the average commoner?
I ask this because when we learn about these ancient empires, it tends to be pretty homogenized. Granted, there may not be a lot of literature on this to prove otherwise but humans are the same throughout history so maybe this happened? "Ah, he's from the South Kingdom." or something. (Another issue could be that the education systems only have so much time to teach.)
So not to be overly pedantic but right off the bat the Roman Empire was not from the Bronze Age, both republic and empire were really the quintessential state from Late Classical Antiquity so quite a few centuries after the bronze age
That being said if you are interested in Rome, as one example, they absolutely did, regional identities were even more important historically than they were today. Even the Romans with their love of citizenship above all else, would give each other nicknames and identifying marks signifying which part of the Empire they were from, perhaps most notable would be "Philip the Arab" so called because he was from Roman Arabia. Nicknames such as Africanus and Germanicus shouldn't be conflated with these however as they were names taken from the locations in which people had scored great victories, rather than from where they were born.
Ordinary Roman freemen and citizens would have some fairly considerable knowledge of other provinces and within Rome itself you could expect to find Greeks, Gauls, Illyrians, Egyptians and many others due to trade from other parts of the Empire, these people were still acknowledged as having differences, partly due to the maintenance of distinct religions and cultures throughout the provinces. The people were aware of each other's idiosyncrasies even if they didn't know the extent of their cultural or religious differences and would struggle to separate stereotypes and rumours from facts. But these people would be aware of those who were of Jewish, Gallic, Greek etc ancestry and refer to them as such.
As for Egypt there were significant cultural differences which changed from which period you're talking about as Egypt lasted for a very long time. A fun fact is that we are closer in time to Cleopatra than she was to the construction of the pyramids but that's besides the point. An interesting phenomena which occurs particularly after the wars of Alexander the Great and the arrival of the Ptolemaic dynasty you see major differences between the traditional Egyptian culture which had existed and evolved over many centuries and the relatively new arrival of the Greeks. The arrival of an entirely new people and culture was new for the Egyptians or at least hadn't happened in a very long time, perhaps since the 25th Dynasty who had originated in the Kingdom of Kush (Sudan). We know that during this period of Greek rule of Egypt, Upper Egypt (with Thebes and Siwa etc) had considerably lower rates of Greek inhabitants than Lower Egypt (With Alexandria, the Pyramids etc). The Greeks ruled as a minority though this should not be likened with Jim Crow or Apartheid South Africa, as Ptolemy saw himself as having divine reasons for ruling rather than racial ones