The classical world wasn't a monolith, and understandings of deities could and did differ from one town to the next. In general, what you thought about the gods was less important than orthopraxy, that is, performing proper ritual. To begin with, though, we can look a little at Greek treatments of foreign religions.

The most common practice was to identify foreign gods with Greek "equivalents", that is interpretatio graecia, but it was common for foreign deities to be discussed by name as well, sometimes due to differing understandings of which Greek deity they corresponded to. Herodotos and others describe the Zoroastrian deity Aúramazda (probably conflated with Elamite Khumban by the Elamo-Persians) as "the sky, which they call Zeus". What this appears to convey is that they considered their chief deity to be celestial, and that they considered it to be nearly all-powerful or vastly superior to other gods. The Greek deity that comes closest to possessing the latter trait is of course the king of the gods himself. Sasanian and Parthian parallell inscriptions suggest that Greek-speaking Zoroastrians also identified Zeus with Aúramazda.

Tacitus' discussion of Jewish religion is a pretty apt example of how discussions of foreign gods could look:

"As their priests sing to the accompaniment of flutes and kettle-drums, and as they deck themselves with laurel, and as a golden vine was found in their Temple, many people believe that they worship Bacchus, the conqueror of the East; but the two cults have nothing in common, for Bacchus has established a brilliant and joyous ritual, while the customs of the Jews are bizarre and morose."

You'd normally expect the god of the Jews to be identified with either Zeus/Jupiter, or perhaps, like El, with Kronos, the father of Zeus. There may be other reasons for the conflation with Bacchus/Dionysus, e.g. that the Phrygian sky-god Sabazios was identified with both Zeus and Dionysus.

There may be attestations of pre-Christian deities being described as purely "local" or "ethnic" by Greek or Roman authors, but none that I can think of. So, I think the direct answer to your question is that the most widely attested view is closer to a kind of universalist understanding of the divine.

For a good in-depth discussion of how the classical world viewed the divine (in particular, a comparison between Homer's and Lucretius' views), which I think will answer any remaining lingering questions, I defer to this answer by /u/DarthPositus .