In Ancient Greece for example you get people like Socrates who's only called Socrates. But in Rome it seemed everyone had 2 names. So when did this happen?
All cultures start with single names. Some countries where that still applies are Iceland, Myanmar, and partially to Indonesia. (There's a library science book on all the different ways Indonesian authors name themselves according to their ethnic group. Some are single-named and some just don't have inherited family names.)
Countries that mandated family names in the 20th century include Thailand, Turkey & Iran.
Now let's establish a little jargon. It's not first names and surnames. It's given names (of the individual) and (inherited) family names.
The oldest family names are from China, where they are the first name in a string. Only the highest classes had them before the 5th century BCE. They can be viewed more as clan names than family names, if it makes any difference.
But the earliest clan and family names are probably Etruscan. As the first culture model for the Romans, the Etruscans bequeathed them a 3-part name: given name, clan name, family name. If you think the Roman tradition was 2 names, it's because later Western authors made the format "more familiar" to their readers.
But wait! It's more complicated. Not all people in Rome had a clan/gens, so they would have only 2 names, low class as it was. Pompey was one of these. His apparent cognomen/3rd name, Magnus, was a title, the Great. His son did not inherit it but was known as Pius because he warred to avenge his father.
Then there were agnomen/honour names, added like medals to victorious generals, that might or might not be inherited.
And then the whole system broke down when the empire fragmented, and was only a dim memory.
In Europe, claims for family names go back to the late mediaeval/early Modern, depending on how you parse that, and how you view one noble family's claim vs hordes of commoners still known by who their father or mother was or what job they do. In the 1880s, Norway still didn't use family names for ordinary farmers. Let's say it was a growing trend, accelerated in the 1800s by autocratic governments wanting more solid ID for taxation and conscription.
And there are Europeans who put the family name first, like Hungarians.
Some Indonesians still have one name until the present .