/r/publiusclodius' answer is correct for the Republican period, but Roman naming conventions changed quite a bit during the imperial period. The praenomen for instance quickly fossilised and became part of the family name, with differentiation between family members being instead achieved through changing the cognomen. It is also true many new citizens took the name of their benefactors, but the rise of "Flavius" in the period you are interested is mostly due to the Constantinian dynasty, which held power between Constantine I and Julian (306-363): Flavius was their family name (gentilicium). This name became widely adopted by the elite, as it denoted status and generally indicated that the person served the imperial government, whether at the centre or in the provinces. Another example would be the name Valerius, which became very popular before Constantine I, as it was the family name of Diocletian, the man who dominated the Tetrarchy. Basically, by this point Flavius had become more of a title than anything else (an analogy might be 'Mr' in modern English) and adopted by those involved in the administration. It wasn't always hereditary, as their sons sometimes took the name "Aurelius" if their careers changed (a name that indicates a lower status, but in fact the most attested to name in late antiquity since there were more of them than the elite Flavii). As your game presumably only dealt with the elite, the fact that most individuals were named Flavius is to be expected. A lot of people still bore more traditional names, but they were often part of the old elite, whereas individuals who rose to prominence under the Constantinian dynasty generally used simply the default Flavius or Aurelius. As for Flavius Arcadius Augustus, Augustus was of course the name used to signify his connection with the first emperor and Flavius the name used to denote authority, Arcadius then was the personal name, hence why we refer to this emperor only as Arcadius. Surnames were rare in the past and they only reemerged in the Roman world in the late eighth century.

If you are interested in a more detailed and nuanced analysis, this article is worth a read, especially as it covers Roman conventions from the beginning to the end. Keenan's classic article, 'The Names Flavius and Aurelius as Status Designations in Later Roman Egypt' in Zeitschrift für Papyrologie und Epigraphik (1973), is also good, as it was effectively through his work that we can understand Flavius as a name indicating status.

That game might not perchance be Attila: Total War? :)

Flavius Arcadius was the son of Flavius Theodosius I. and Aelia Flavia Flaccilla, his brother was Flavius Honorius. Augustus was his title as Emperor of the Eastern Roman Empire, not his name (Augustus oscillates a bit between the poles of 'name' and 'title', so take this with a grain of salt) - his full imperial title was Imperator Caesar Flavius Arcadius Augustus. It's true, all emperors following Augustus took on this title (which was itself bestowed upon Augustus in January of 27 BC). From the 2nd century on, the title of Augustus was also given to junior emperors, like Lucius Verus to Marcus Aurelius, or Maximian to Diokletian.

Flavius was his nomen gentile, the name of his gens, or his family name, if you will. Flavius is a pretty popular name for several reasons. It was an ancient Roman gens, of plebeian origin, who during the Republic had tribunes and consuls among them. Then there was the dynasty of Flavian emperors, Vespasian, Titus, and Domitian. In the time of this dynasty also fell the reason for the big popularity of the name: When peregrines (inhabitants of the Empire without full Roman citizenship) attained citizenship, by imperial grant, military or civic service, or by emancipation, they usually took on the nomen gentile of the reigning emperor, so there are many Flavii, Ulpii, Aelii, Aurelii, Iulii or Cocceii; these gentile names are therefore also often called imperial gentilice, Kaisergentilizien in German. These names could then also be perpetuated by the offspring, leading to an ever increasing number of people bearing this name.

Constantius Chlorus, founded as Flavius Valerius Constantius what is often called the '2nd Flavian dynasty', and from Constantin I. on all following emperors assumed the name Flavius when they came into office. So from the 4th century on, the name was adopted by usurpers of the imperial throne as well, who thus changed their name (Valerius was another popular name to gain legitimacy). Members of the imperial bureaucracy and military commanders in Late Antiquity also took on this name as a sign of their office and their elite status, like the magister militum Flavius Aetius, or Flavius Felix. Since in Total War, all the people you deal with on a strategic level would be members of the absolute elite, it makes perfect sense that they are all named Flavius in one flavour or the other.

Flavius Constantius thusly assumed this name as magister militum, and so did his son, emperor Valentinianus I., his son-in-law* Theodosius I., and that one was the father of emperor Flavius Arcadius.

It is a general trend of Late Antiquity that the nomen gentile shifts from indicating familiary relationships and became more of a marker of social status. Roman naming practice was essentially for most people reduced to a single-name system, which is why he is usually known as Arcadius. Flavius was almost more of a marker than a name.

*see below

The traditional Roman male name by the time of the late Republic has three parts: Praenomen (kind of like a first name, except with a very limited list to choose from), the nomen, which was the gens ("clan") name and cognomen, a more specific nickname of sorts that was passed down to other members of that family. So, for Gaius Julius Caesar, Gaius= praenomen, Julius= nomen, and Caesar= cognomen. His name was Gaius and he belonged to the larger Julian clan and the Caesar subdivision of that clan. In private contexts, some friends or family might have called him Gaius, but generally he would be known as Caesar or by his full name, and nobody would have ever called him Julius. Publius Cornelius Scipio Africanus had two cognomina: "Scipio" which he inherited, and Africanus which he personally obtained through his victory over Carthage. His son inherited both.

Things get more complicated, though. Most non-aristocrats did not have a cognomen, and even some elites didn't: For example, Gaius Marius. And, to get to your main question, slaves, once they were freed, adopted the gens nomen of their former master. Sulla provides an example. In the late 80s B.C., to quote Appian (B.C. 1.100), "to the plebeians he added more than 10,000 slaves of proscribed persons, choosing the youngest and strongest, to whom he gave freedom and Roman citizenship, and he called them Cornelii after himself." So, when you start having powerful individuals who can free many, many slaves (such as emperors), you're going to start having lots of freedmen with the same clan names. And there seem to have been many, many Romans with the nomen Flavius in the late first century A.D. and second century A.D., so scholars have assumed that the emperors of the Flavian dynasty (Vespasian, Titus and Domitian) were responsible for freeing many slaves. Over time the distinction between praenomen and nomen broke down, and Flavius almost became a praenomen itself.

As for someone named Flavius Arcadius Augustus, I'm pretty skeptical. Augustus was a cognomen reserved for emperors. That said, it did become more and more acceptable for non-elites to adopt cognomina themselves during the imperial period, and by the late empire you get more and more interesting names. My personal favorite is the emperor Maximianus, whose full name was Flavius Clemens Magnus Maximus Augustus (my very loose translation: "Flavius the Forgiving Great Greatest Emperor"). And it's always worth keeping in mind that Roman name conventions are constantly evolving.