As the title states, it has been my education that the Romans would explain other gods as being different versions of their own. One in particular I can think of is Hermes, i.e. Mercury, being Odin in the Norse mythos. I was also under the impression that this sentiment continued into the transition over to Christianity with the cult-like worship of saints and the claiming of sainthood of many individuals being the cause of important historical events.
Why did this mentality of seeing other cultures dirties as other names for gods or saints stop? I always thought it was quite a useful tactic to bridge religious disagreements. Did this stop during Rome's fall? Or did it stop earlier. I have tried searching around for information, but it seems to be a difficult subject past the transition to Christianity in Rome.
What you're referring to is what is termed syncretism. Roman syncretism evolved quite a lot over the period of the Republic and then into the Empire, but in general, Romans were happy to leave local deities and practices in place so long as they were compatible with their own. This was usually not an issue for polytheistic cults, and gradually, starting in the 2nd century, when the Empire became less and less focused on the actual Roman people (in Italy) and increasingly ruled by emperors drawn from all over its territories, it became even more prevalent. It was a problem for strict monotheistic religions, primarily because of the conflict with the Roman Imperial cult (where emperors, both living and deceased, were viewed as divine beings). The Romans saw the inability to acknowledge the imperial cult as a sign of disloyalty, which is one of the main reasons the Jews had so many conflicts with the Romans through the first and second centuries, resulting in the destruction of the Temple in 70 AD and the Jewish diaspora after the bar Kochba revolt under Hadrian.
Once the rulers of Rome adopt Christianity, the role of the Emperor shifts from divine being in his own right to protector and patron--and ruler-- of the Church. The state-sponsored imperial cult disappears, and imperial patronage means that gradually, pagan cults of all kinds within the bounds of the Empire begin to decline, their worship sites sometimes taken over by Christians. But the decline of the Western empire itself meant that when the Germanic successor states converted, they did not do so via conquest, but because their leaders converted (usually encouraged by local bishops or missionaries), which meant, from a practical sense, older religions and Christianity often existed side-by-side for years. Religious practices in agricultural areas are often highly conservative--farmers believed the success of their crops depended on them--and so it makes a lot of sense that the practices would continue but under the patronage of a local saint (or Our Lady if there weren't one). So syncretism itself also evolves.
A great example of this is St. Nicholas, everyone's favourite heretic-punching bishop. I recommend Saint Nicholas of Myra, Bari, and Manhattan by Charles W. Jones if you want to learn more. St. Nicholas took on attributes of Poseidon/Neptune (hence he is a patron of sailors) as well as Wodan/Odin (and to some extent, Thor) . As I think you'll see, syncretism persists well into the medieval era, but the difference being that it's far less often a planned approach by the Church hierarchy than an evolution of popular practice.
And syncretism does persist into the modern era, where missionaries typically explained Christianity in terms local populations would understand. (I'm thinking right now of the "Huron Carol" by Jean de Brebeuf, with its original Wendat lyrics using imagery and terminology that would have been familiar to the Indigenous peoples he was evangelizing.) By then, of course, syncretism is a conscious approach to converting populations to Christianity.