The crux of my question is whether people generally believed that only their own gods existed, or if they accepted the discovery of new gods when encountering foreign tribes.
Ancient history has many recorded incidents of foreign gods, temples and other symbols of religion being disrespected, desecrated or destroyed entirely during conflict. Were people afraid of repercussions from gods they had newly encountered but nevertheless disrespected?
What was the intent behind these kinds of actions?
To deliberately disrespect those gods, because they believed them to be real but 'allied with' their enemies?
To simply offend the people or eradicate their religion, with no belief in their gods?
Did people believe that their own gods would protect them against foreign gods that might be angry?
Worth mentioning when discussing this topic is a historic phenomenon that, for example, occurred in the Roman Empire and among surrounding peoples that is known as interpretatio. When the Romans would encounter the gods of others, they would often interpret them as forms of otherwise extension of their own gods rather than dismiss them as 'false' or 'inventions'.
An excellent example of this phenomenon is when the ancient Germanic peoples and the Romans encountered one another, they actually both did this. In turn, in sources like Tacitus's Germania (in fact, I believe the phrase interpretatio romana first appears here) and on Roman era inscriptions in regions where the ancient Germanic peoples and Romans intermixed, you can find what scholars generally accept as early Germanic Odin referred to as Mercury, and Thor referred to as either Jupiter or Hercules. This process is known to scholars as interpretatio romana. (Of course, as Tacitus describes, this did not stop the Romans from doings like burning sacred groves dedicated to the deities of others.)
The ancient Germanic peoples appear to have done this in reverse, a process known as interpretatio germanica. This produced most of the weekday names of today's Germanic languages. This is why, rather than Mercury's Day (that is, a direct extension of Latin diēs Mercuriī), we have 'Woden's Day', modern English Wednesday (Wōden is the Old English form of the deity's name and it ultimately stems from the same early Germanic source as the common modern English form Odin, which English borrowed from Old Norse Óðinn).
It's the same case for every English day of the week (with the exception of Saturday in the English language, which for some unknown reason was not subject to interpretatio germanica among English weekdays).
You can read more about this in handbooks on the topic, such as:
* Lindow, John. 2002. Handbook of Norse Mythology. Oxford University Press.
* Simek, Rudolf. 1996. Dictionary of Northern Mythology. D. S. Brewer.
Of course, interpretatio didn't always happen. Sometimes deities are simply absorbed into a pantheon, which we see in the case of Isis.
I'm most familiar with Judaism, but it's also particularly instructive because the theology transitions between a polytheistic cosmology and monotheistic cosmology over the 1500ish years before the common era. This is to say "this is our God and we like it the best, we have a covenant with that one only" and "this is the only god that exists and yours don't exist or are really just aspects of our God." The first commandment says you can't put any god before God, but doesn't say you might not be able to put some after him. elements of this are retained in the bible in early books. In the bible, the narrative can also be understood as contests between gods to outright denial of other gods. The difference between "my god is better than your god" and "my god is real and your god isn't" is I think what you're getting at.
I'm getting a lot of this from Paul Johnson's A History of the Jews
So to answer your question, ancient peoples frequently acknowledged that other gods existed. They often adopted them alongside their own, the spread of the cult of Isis in the Roman empire alongside the latin pantheon. The imperial roman cult (the deification of the emperor) specifically never negated any other gods.
It's important to remember that many of these religions never had anyone who could say "this is true or this is heresy" there was no religious structure for that. and that the 'pantheon' is a pretty new idea to analyse ancient religions as discrete and internally coherent, which historically were quite muddy. For example, the ancient Egyptian and Greek pantheons completely collapsed into eachother when Alexander conquered Egypt. The Ptolomy's blended them on purpose to enhance their legitimacy.
So when it comes to war and desecration, what you're doing is a test. if I destroy a temple on behalf of one god, and then I win the battle, well my god was stronger. Or, I'm literally killing that god, because that's not a stone representation of a god, that literally is the god, and the temple literally is where that god lives. The sort of ethereal incorporeal radical cosmological unity of the late Jewish or Christian God also can't be taken for granted.