Native Roman religion was quite distinctive before it assimilated Etruscan and Greek elements. There were gods of forests, hearths, fire, pantries, children suckled by wolves, etc. Lots of it belonged intrinsically to the cultures that lived along the Po that would eventually become Rome.
But as Rome grew in power and became an urbanized military fortress of power, it assimilated religious values. The Etruscan civilzation had died out, the golden age of independent Greece was over, so Rome took their traditions upon itself. Keep in mind that they were all highly regarded as fantastic civilizations by Rome. It was an honour for a Roman noble to claim Etruscan descent, and for Romans to know Greek philosophy and arts. Their religious elements were no different.The Romans admired the structured Olympian pantheon the Greeks revered, and adopted it for themselves, because it gave order to the universe, complimenting the order the Romans were establishing on earth. Aeneas, the forefather of the founders of Rome, was a Trojan who survived the War to settle in Italy, as chronicled in the Aeneid. So the Romans claimed descent from Troy, a popular city from the Greek Iliad, as well.
But the fact that both cultures ultimately fell to Rome was the linchpin. Romans believed they had perfected the traditions of these failed foreign nations by making them a part of its civilization. The Vestal Virgin priesthood, for instance, was created by a combination of the assimilation of the Greek goddess Hestia after Rome came to power, and the tradition of a woman tending the spirit of a hearth fire before Rome came to power. The cults of Isis from Egypt and Mithras from Persia became very popular in Rome. Serapis, a synthesis of the Egyptian gods Osiris (Wesir/Asar) and Apis during the Greek rule of Egypt, also flourished in Rome.
The Romans knew the power of mythology probably better than anyone in the Western world. They knew from their own experience that religion can shape political power over a nation as much as military strength. Everywhere they went, not only did they absorb some traditions of the cultures they encountered, they also spread their own influence. Roman influence is very visible in Celtic nations, such as the springs of the healing goddess Sulis Minerva in Bath, England. If you go by past history, the Roman Minerva, inspired by Etruscan Menvra, then Greek Athena, and possibly the Minoan Athina Potinia, would have been responsible for this synthesis. The Romans knew this and were simply perpetuating the cycle with the Celts accepting Minerva alongside their Sulis.
As people have already pointed out, the connections between gods wasn't always so clear cut and varied according to the social attitudes of cultures contrasted to Rome's.
It's because of Romans' high regard for cultural assimilation that Classical, Egyptian and Celtic myth survived through history to today. Even the Christians followed this when it turned some Celtic figures into saints, preserving their pagan memory while attempting to stamp out their pagan influence.