Fame in that sense has been a pretty constant thing throughout any part of history where you have large settlements of people and a prosperous enough economy to maintain a class of entertainers. Romans certainly had favorite Gladiators who were known by name and who even endorsed products and services. We can assume from our existing records of Roman wall writing (such as the wonderfully funny Pompeii inscriptions) that most towns would have had well known people who were the equivalent of modern famous Twitter or Facebook accounts. The Pompeii writings contain numerous "posts" about how good or bad so and so is at poetry or how attractive various prostitutes are, etc. When you get later into history one of the first things mass production art and photos were used for was making trading card esque images that people could swap around. As acting became a more and more prestigious occupation between the middle ages and now actors became more and more well known and important. John Wilkes Booth for instance was an extremely well known and popular actor who people would travel to see in various things because they were big fans. In the days of dime novels and other mass produced stories, heavily embellished or made up tales about technically real people were common as well. Especially stories about pioneers and contemporary cowboys/adventurers. The problem with documenting this kind of stuff is that much like today I imagine it was fleeting and shifted with trends. Those are the kinds of things people often forget to commit to the greater canon of history in favor of things like kings and generals. We just have the ability to display these sorts of things on a wider scale. A pretty lady can have her instagram posts reach literal millions of people, back in ancient Rome it would have been limited to a bunch of local dudes scratching "HAVE YOU SEE THIS LADY SHE SO HOT" onto the walls of alleyways.