They're just labels, entirely meaningless.
Usually the "primary function" is still used as a label, or the "most important function", so probably genitive. But that's not an analysis.
Cases merge all the time, and we apply one label or the other. "Accusative" in English includes everything that isn't nominative. But it's just a name. (Some people care about this enough to use the term "oblique", though.)
As a simple example:
Latin: Nom, Acc, Gen, Dat, Abl
German: Nom, Acc, Gen, Dat
Where Latin uses the Abl usually German uses Dat. But it's called "dative".
There's no theoretical motivation for this label, nor should there be. It's just a name.
Also, cross-linguistically case names are applied very strangely, sometimes for almost no reason at all. And, yeah, it's biased by Indo-European. But again, they're just names. As evidence of this, look at Arabic where Nom and Acc are relatively similar (subjects and objects, but maaaany other functions too, such as Acc being used for adverbs), while Gen is used as the label for a case whose primary usage is not for possession.
