Depends on how you interpret "Real" and "False".

The Greeks and Romans tended to view the foreign gods as manifestations of the same deities.
I.E. Amun was Zeus, obviously, they just had a different interpretation of Zeus. (This also gives us hybrid gods, like Amun-Zeus) Although sometimes they'd adopt the new god into their pantheon (Cybelle was not Greek in origin for example, and Amun also got worshipped as just straight up Amun IIRC) Pliny the Elder in Natural Histories describes it with the phrase "Nomina alia aliis gentibus", or in English "Different names for different peoples".

But it's not just different names of course, they also serve different roles (Most famously Mars is one of the principle Roman gods, while Greek writers seem to treat Ares with decidedly less respect.) and some even serve multiple roles (I.E. a Celtic god may be aspects of many Greek and Roman gods), but all of that is allowed within the Greek and Roman framework of understanding the gods. Because even within their own space, the gods had multiplicities of roles and manifestations. Jupiter isn't just Jupiter, there's multiple understanding of what Jupiter IS and those can be contradictory or overlapping with other gods.

The concept is known as "Interpretatio Graeca", or Romana for the Romans.

There's actually a really interesting bit about this happening with Germanic deities in The Fall of Rome: And the End of Civilization by Bryan Ward Perkins.

It's worth remembering that neither all ancient Greeks nor all ancient Romans necessarily held the same sets of beliefs as one another to start with. These were big areas of land with lots of individual states/cities/communities whose religious practices weren't based on a big instruction book on what to think. As a result, Greek and Roman religions had always found ways of managing the co-existence of different, even contradictory beliefs.

One way of doing this was to use the obvious parallels/overlaps that u/somedudeorwhatevs mentions to help people work within a shared framework of understanding. This was especially effectively used by Romans in the earlier empire. Other shared mythology seems to have helped tie the various different versions of the same gods together (like Hesiod and Homer, for example).

But I think it's difficult to imagine how important it was that the mindset was just there from the start. When a Roman encountered a new cult, for example, they could pretty easily accept it as just as real as their own beliefs, without necessarily taking it into their personal religious practices. In the absence of dogma or orthodoxy, as long as you were seen to be generally doing the right thing (you can call this orthopraxy), it was probably ok. You can see this in how quickly some of the oriental cults (like Isis in particular) were taken on in Rome. It didn't involve a conversion so much as an addition. My view pretty much chimes with Charles King's 'The Organisation of Roman Religious Beliefs' here - a very worthwhile read on a fascinating topic.