Historically, how did Esav come to be associated with Rome/Christianity? How did Yishma'el come to be associated with Arabs/Islam?
According to Ramban, descendants of Eisav, became followers of Jesus. They migrated to a land called Edom, near Italy and were initially persecuted there. Eventually they and their religion were embraced by the Romans, who were really descendants of Yavan (Italia shel Yavan). Hence the two peoples merged and Edom and Rome became one.
The Arabs were already associated with Yishmael at the time that Onkelos the Convert wrote his translation of the Torah. Although the same association is used in Targum Yonason Ben Uziel who lived two generations earlier (maybe even three!) he may not be the actual author of that translation and it may actually date from some time after the founding of Islam. See Parshas VaYeshev 37:25.
Besides for the Ramban cited by @Yahu The Ibn Ezra attempts to explain this as well (Bereishit 27:40)
Rome, which led us away captive, is of the seed of Kittim; and so the Targumist has said, in Num. 24:24. 'And ships shall come from the coast of Kittim.' And this is the same as the Greek monarchy, as I have explained in the Book of Daniel, and there were very few who believed on the man of whom they made a God. But when Rome believed, in the days of Constantine, who changed the whole religion, and put an Image of that man upon his standard, there were none in the world who observed the New Law, except a few Edomites, therefore Rome is called the Kingdom of Edom.
According to the Ibn Ezra Edom and Rome are synonymous since they both promoted Christianity, and the Edomites were the first to accept the religion.
The Radak however explains this differently (Yoel 4:19),
The prophet mentions Egypt and Edom; Egypt on account of the Turks, and Edom on account of the Roman Empire; and these two have now had the dominion for a long time, and will continue until the redemption. This is the fourth Beast in the vision of Daniel. And this is said because the majority is composed of Edomites. For although many other nations are mixed among them, as is also the case with the Turkish Empire, they are called from their root.