In popular culture, polytheistic gods are presented as "The god of _______." How accurate is this and how applicible is it to different ancient pantheons?

Oh hey, something I’ve done research on! I will try to answer this from the Roman perspective.

It’s true that Roman gods, to some extent, had certain domains with which they were associated. However, you’re right to be skeptical that it’s as simple as saying that Jupiter was the “god of thunder”, Juno the “goddess of marriage”, Pluto the “god of the dead”, and so on.

Jupiter did indeed have powers over the sky, but this is just one aspect of what made Jupiter, Jupiter. Another aspect we might look to is the extent to which his worship, and the symbols involved, were seen as relating to the power of the Roman state. The chief priest (“pontifex maximus”, a title which has since been claimed by centuries of Popes to the modern day) was a priest of Jupiter. The scepter (sceptrum), a symbol of power, was associated with Jupiter (Festus, De verborum significatione, 210). When victorious Roman generals led their army in a procession (a triumph) through the city, they wore purple regalia and their faces painted red to embody Jupiter, and their parade ended with them going to the Temple of Jupiter Optimus Maximus (“Jupiter the Best and Greatest”) on the Capitoline Hill, a temple he shared with his wife, Juno, and his daughter, Minerva. When consuls (the two chief magistrates) came into office every year, they sacrificed an ox to Jupiter and his will could be discerned through the body parts of the sacrifice – such as the liver, which Livy mentions as having vanished from one year’s ox, a bad omen (Livy, Ab urbe condita, 41.14.7).

From these we can see that a large part of Jupiter’s function within the Roman pantheon was as the king of the gods, who could grant legitimacy to the Roman state. In short, to call Jupiter a “god of thunder” is technically not incorrect, but is quite an oversimplification that misses the nuances of function and meaning associated with his cult in the context of Roman public religion. If you really wanted to simplify things, you would probably be better served thinking of Jupiter as the god who represented the legitimate power of the Roman state rather than as the “god of thunder”. This better reflects his function as a public cult deity.

On top of this, another characteristic of Roman religion was the idea of gods having different aspects. Jupiter, being a rather powerful and important god, was worshipped all over the empire, but in different towns (and even within the city of Rome itself) one might find multiple versions of Jupiter. This can also be said of other public cult deities, such as Mars. Augustus had a temple dedicated to him, or more specifically to Mars Ultor (“Mars the Avenger”) no doubt as a message to the Roman public that he had avenged his adoptive father Caesar’s death at the hands of Brutus and his friends. Also in Rome is the Temple of Venus Genetrix (“Venus the Founder”, i.e., of the Julian family; the Julii claimed to be descended from Venus). The different aspects of gods were rather useful for imperial propaganda.

Going back to Jupiter himself, there was also a Temple of Jupiter Stator (“Jupiter the Sustainer”) in the Campus Martius (“Field of Mars”, in the northwest part of the city). His wife, Juno, in addition to being represented in the Temple of Jupiter Optimus Maximus, could also be found as Juno Moneta (“Juno the Warner”) guarding the state’s funds in her temple, also on the Capitoline. (The association between Juno Moneta and the state treasury is a likely explanation for the modern day “monetary”, “money”, etc.).

Though ostensibly the same god, one temple in one part of Rome, or in another city of the empire, could house within it the worship of Jupiter, Juno, or any other public cult deity in one of many different guises. In fact, some historians have argued that public religion was so tied to civic identity (in other words, the worship of the gods in localized forms was so tied to each individual city or town) that the attempts of emperors like Decius in the 3rd century AD to enforce a more coherent and consistent religious identity were doomed to failure (Rives 1995). It’s worth noting that a fundamental characteristic of ancient religion was its diversity. Indeed, it’s become common among historians of Roman religion to speak not of “a Roman religion” but of plural “Roman religions” which were never really brought together until the rise of Christianity, which concerns itself much more with orthodoxy.
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