I actually answered a thread on how religious practices varied among the Greek city-states here and a separate on the syncretization of polytheistic religions here and I have copy pasted it below for relevancy:

This is a phenomena called interpretatio Graeca by modern historians, where Greek observers used their own beliefs and legendary traditions to interpret and understand foreign deities and myths, and these cultures in turn did the same. This is apparent in the synchronization of Greek, Etruscan and Roman deites\heroes such as Zeus\Jupiter, Ares\Mars, Hera\Juno Heracles\Hercules etc.

This was especially prevalent during the Hellenistic period, when Hellenic and Philhellenic culture was prevalent even where there was not a ruling Greek or Greek-speaking class such as the Georgian Kingdoms. Syncretic religions emerged such as the Greco-Egyptian religion in the Ptolemaic Kingdom (Hermanubis, Zeus -Osiris, Hera-Isis, Dionysius-Osiris, Hermes-Thoth, Horus-Apollo, Bastet-Artemis, Ptah-Hephaestus). The cult of Isis in particular was quite popular and was worshipped by an apparently large amount of people in Rome by the Late Republic because a temple of Isis was planned and a section of city reserved for it. And throughout the Hellenistic era there was significant cultural and religious exchange and synchronization between Bactrian\Indian, Persian, Near Eastern, Berber and even to an extent Celtic traditions. Aphrodite was equated with Astarte, Ishtar, Venus and Isis for example.

The Romans also adopted this method of interpretation (interpretatio Romana) and you can see this with the emergence of a distinct Gallo-Roman religion including roman deities adopting Gallic epithets and syncretic Gallo-Roman practices emerging (Mercury-Wotan was the principal if memory serves but Gallo-Roman culture is not my forte to the extent Hellenistic culture is, perhaps someone with a sharper focus on the subject can expand on it).

Some reasons this was so prevalent in ancient times is because deities often had multiple, sometimes contradictory natures and polytheism lent itself well to the kind of cross-cultural adaptations that occurred.

Another interesting thing to note is that it is theorised that many of these religious pantheons are all rooted in the same Proto-Indo-European mythology, which had a pantheon based on a human family (Zeus-type = Father, Hera-Queen of Heaven type = Mother etc.) and that most deities map to certain functions, (ie. Sky-God, Divine-Twins, Three Fate Godesses etc.) as well as recurring myths such as Divine Son, Sun\Moon siblings, as well as similar Creation and Flood myths (the Egyptian creation story in particular, although not Indo-European, reminds me in certain ways of traditional Abrahamic creation story, although obviously there are differences due to one being distinctly polytheistic and the other distinctly monotheistic).

Christianity also absorbed elements of local religious practices as it spread which can still be seen today (ie. images of Mary and Jesus being related to Isis and Horus, Lake of Fire motif from Egyptian mythology, Roman and Gallic holidays being adopted in altered forms, and traditional beliefs about certain folk knowledge and practices remaining prevalent in Christian communities).

However, not all deities, myths and practices found their equivalent across the board; Janus and Terminus are both Roman deities with no Greek equivalent, and Apollo was directly transferred into Roman religion from Greek tradition, and many Greek deities and cultic practices have conflicting and distinct regional traditions and importance dating back to the Minoan period. Not to mention the ways that Near Eastern, Mesopotamian and Berber deities both remained separate from and influenced religious practices in the Mediterranean.

Sources:

Germania by Tacitus

Natural History by Pliny

Celtic Culture: A Historical Encyclopedia by John Koch

Encyclopedia of Indo-European Culture by J.P. Mallory and Douglas Adams

The Histories by Herodotus

Parallel Lives by Plutarch

Moralia by Plutarch

Iliad by Homer

Odyssey by Homer

But essentially, no they did not usually perceive themselves as being the "true" religion in a more modern sense, because polytheism in the ancient world tended to be more cultic than dogmatic, and polytheism lends itself to multiplicity. Your comparison of Greek patron deities is an excellent one because Greek religion was less of a standardized religion and more of a collective tradition of practices and beliefs that first emerged and started to evolve during the Minoan period or perhaps earlier. Patron deities were universal deities but they might be especially predisposed to certain areas due to that being their birthplace or because they are related to that region for some reason in their mythology, foreign deities were often adopted by armies serving in the regions they are native to, and during the Hellenistic period in which much of the Mediterranean and North Africa had been previously conquered and Hellenized by Alexander the Great, the surge in the popularity of the Hellenized version of the Egyptian religion (particularly the cult of Isis) and in the Roman Republic Greek deities were often conflated with existing Roman\Etruscan deities, or outright adopted into the Roman pantheon (as with Apollo being adopted as Pheobus Apollo).

In this way, Abrahamic religions were fairly unique, at least in the Mediterranean.