As Apollodorosh did, it’s important to start by distinguishing between eclecticism and syncretism. The difference may be subtle, but nonetheless important. The former is the combining of different elements from different backgrounds without creating a unified system; it’s unsystematized diversity, if you will. The latter refers also to a combination of diverse parts, but into a systematized whole, where contradictions have been dealt with and unified forms created. So, for instance, worshiping Zeus and Amon side by side as separate gods would be eclecticism, while morphing Them into a single form with its own system would be syncretism. A related term is monism, which is the belief in a common source from which spring different manifestations.
The theological question
Are gods from different communities, but with shared functions, different and individual deities or cultural manifestations of the same entities? This question is not easily answered, even when historical information may allow one to track down the origins of a god to another deity, because ultimately it’s a matter of personal faith and gnosis. As such, what you’ll read here are solely my beliefs and will not necessarily be shared by other polytheists, Romans or otherwise. The following examples can illustrate both the question and my insights on it.
Are Zeus and Jupiter the same god? There’s a tendency to syncretise Roman and Greek deities and not without historical basis, but personally I tend to be a sceptic and take Them at face value, as separate entities, unless I get clear signs or experiences of otherwise. Of course, the fact that They’re both thunder deities at the head of neighbouring Mediterranean pantheons may indicate that They’re the same, but then I wonder if we’re not dealing with two individual thunder numens, one from Mount Olympus and another from Capitoline Hill, both “promoted” in time to the highest status in Greek and Roman pantheons, respectively. This latter possibility would naturally mean that Zeus and Jupiter are neither thunder itself nor do They have exclusive control over it, which may generate new questions.
Minerva and Athena are another good example. The former seems have originally been an Etruscan goddess that naturally found Her way to the Roman pantheon in an early period and by then She may have already been depicted like Athena: with a war helmet, a spear, a shield, and/or an owl. However, identical guise doesn’t necessarily indicate the same goddess, as it’s not always easy to distinguish between the usage of one deity’s outer look to represent another and outright syncretism or importing. Was the Etruscan Menerva an Italian divinity depicted like Athena or Athena as worshiped by the Etruscans? I used to be sure of the latter, but not anymore.
The case of Hermes and Mercury may be clearer. Since the latter had no flamen in Rome, it is assumed that He’s not an early Roman deity, but there are news of a temple to Him dedicated on the Ides of May of 495 BCE. One theory is that He’s Hermes imported by Roman merchants from the Greek cities in southern Italy, which would fit His name: Mercurius from Latin merx (merchandise, business) and mercor (to buy, to trade). In other words, a foreign god spotted by traders who then took and named Him after their job as their patron.
Finally, there’s the case of greater deities of major bodies, namely the sun, the moon, and the Earth Herself. Is the sun Amon, Helios, Sol, Sunna, Amaterasu, or another god/ddess? This is where I tend to take a monistic approach and believe that the spirit of these unique and greater bodies has different avatars, sometimes within the same culture. The numen of the Earth manifests itself as Cybele, Gaia, Geb, Nerthus, and Jord, to name a few examples, which are all individual and autonomous faces of a single entity. A different case is that of deities who work the soil (like Freyr and Demeter, for instance) and local wights who inhabit rocks, hills, or other natural places: these, I’d say, are independent beings, not avatars of greater numens. It’s like a distinction between the planet itself, people who work the fields (individual, despite a common occupation), and those who live on the planet (which don’t necessarily have to be farmers).
Hopefully, I expressed myself in a comprehensible manner (especially on this last part). The question to tackle next will be on the nature of syncretised deities, namely those that combine gods who developed separately for thousands of years.