Not just wealthy Romans--legally all women only had one name, the feminine form of the name of her gens. This isn't a hard-and-fast rule, though--we've got things like Clodia Metelli, whose second name is the genitive of her husband's--and during the Principate it became somewhat popular for upper-class women to give themselves cognomina. But no, women did not have personal names, praenomina, the way men did, and they usually did not inherit the cognomen of their branch of the family. But in Roman naming conventions, neither of those are very important--cognomina only serve to identify specific branches of the same gens and praenomina are more or less irrelevant and totally useless. Latin only has a handful of praenomina (like, around two dozen or so)--this, combined with the fact that particular families often used the same praenomina over and over as a mark of traditional distinction, meant that there were dozens of boys with exactly the same name running around the city. If we add in generations it's much more complicated--the number of Publius Claudius' in Roman history is nothing short of absurd. In the same way that cognomina only identified a branch of a family (for which you needed to know the family to know who the individuals therein were), a praenomen only indicates an individual to his close associates. Simply knowing the praenomen of a person tells you nothing, unless you already know him--praenomina were only used among close associates, normally you'd just call someone by his nomen. That's important--in Roman naming conventions the nomen is the only really important legal name, the others all just identify an individual more concretely. In the video's example, the important part of Lucius Domitius Ahenobarbus' name is the Domitius part--usually he'd just be referred to as Domitius Ahenobarbus, or possibly Lucius Ahenobarbus if his clan was well-known and large (as the Ahenobarbi were). And the standard, go-to way that anyone in the city would address him was simply as "Domitius" more often than not

The video, however, is totally wrong when it claims that little Domitia's dad would've chosen to name her and all her siblings Domitia. That's not a choice at all, it's a legal convention--legally that's the only name that Domitia can possibly have, in the same way that her brother's legal name rests on the masculine form of his nomen. There were, however, many ways to distinguish individual women. The simplest was just to call the girls Prima, Secunda, Tertia, etc. Another was to distinguish them by physical characteristics--"tall Domitia," "dark Domitia," "little Domitia" (Domitilla--this method is quite common, although usually it's used to describe a granddaughter while her grandmother of the same name is still alive). The video is also misleading when it claims that the Romans couldn't figure out which sisters were married to whom, that's simply untrue. Otherwise Clodia Metelli's name doesn't make sense--she's literally, "Metellus' Clodia," i.e. the wife of Metellus. If that doesn't tell us whose wife she is I don't know what does--this isn't particularly uncommon. Although legally women only had a single name, the feminine form of their nomen, there were lots and lots of unofficial ways to distinguish between individuals. Also, numbering children is hardly restricted to women. Many of the most popular and common praenomina are simply masculine numbers--Quintus, Sextus, and Septimus are all just "Fifth, Sixth, and Seventh." There's nothing strange about numerical names in Latin, although technically Prima, Secunda, etc. are not formally names, since they have no legal authority