Many ancient people definitely had fluid ideas about divine identity. The Greeks and Romans are most famous for this kind of thing through the interpretatio graeca and interpretatio romana whereby dieties of other cultures were described through their Greco-Roman equivalents. Indeed, the equivalence between Greek and Roman dieties themselves is a product of this kind of thinking... the Roman Mars and Greek Ares are "originally" quite different sorts of gods, but became conflated through cultural exchange.

One of our very few sources for understanding ancient celtic religion is Julius Caesar's report of the war against the Gauls. It has many problems as a source, one of which is that he insistently refers to Gaulic deities by their "Roman names" calling Mercury the most important Gaulic god.

(Incidentally, see the days of the week for a modern example of the interpretatio comparing Germanic and Roman gods... Moon=Luna, Tyr=Mars, Wodan=Mercury, Jupiter=Thor, Frigga=Venus.)

But this process also took place outside the greco-roman context, as for instance in the identification of Seth and Baal Hadad in the Egyptian context. In many cases, it is difficult to determine when two different gods were identified, and when worship of one particular god was imported and translated for a new context. Sometimes, both happened, where understandings of an originally single god diverged to a great extent, but when trade or migration brought divergent cults together, they interacted and identified their gods with one another.

The Hebrew Bible is an idosyncratic but nonetheless interesting source for the identification of various regional gods. Part of how YHWH worship worked at a certain point was to identify different gods as all being YHWH. YHWH took on the names and honorifics of other gods at this stage, unlike in later stages where YHWH was only imagined as usurping their powers, not their identities.

One classic example is from the book of Exodus (6:2-3) "God [Elohim] also spoke to Moses and said to him: “I am YHWH. I appeared to Abraham, Isaac, and Jacob as El Shaddai, but by my name ‘The Lord’ YHWH I did not make myself known to them." The originally distinct gods Yahweh and El Shaddai are being identified, but the name Yahweh is being privileged as the "true" name. Plenty of other divine names from El Elyon to Baal (!) are used of the god of the Hebrews in the Bible.