The Edomites were an ancient Semitic people believed to be descended from Jacob’s (Israel’s) brother Esau, making them the closest relatives of the Israelites. They were later called Idumeans, just as the Judahites were called Judeans. They were forcibly converted to Judaism late in the pre-Christian Second Temple period. The kings and other leaders of the Herodian Dynasty were Jews of Idumean (Edomite) ancestry. After the destruction of Jerusalem and the expulsion of most Jews from the Holy Land the Idumeans disappear from history, likely being absorbed into general Jewish culture and forgetting their Edomite origins.
Later Judaism associated Rome with the Edomites, likely due to the political connection between the Romans and Herodians, and of course as a way of introducing Rome into Biblical prophesies that did not mention it, by making the Edomites, who were no longer a threat, represent the new enemy- Rome. It’s not surprising to me that some would later extend this to Catholics due to the importance of Rome in Catholicism, and because of the partially real and partially imagined grievances the Jews have with Christians.
However, the more Biblical (and, coincidentally, historical) connection would be to the Jews themselves.
The theme of the younger brother taking over the inheritance of the older is common in the Old Testament. It happened with Ishmael and Isaac, Esau and Jacob, Ephraim and Manasseh, and, in a more fuzzy way, with David and his many older brothers, and perhaps other things like the Davidic dynasty vs. Saul or later Solomon not being the oldest son of David. One might even draw a parallel with the Maccabees, who though descendants of Aaron were not of the family of priests that were supposed to be High Priests, yet for those of us who accept 1 and 2 Maccabees as Biblical they seem to have been chosen by God to lead the Israelites.
Though perhaps not 100% politically correct today, the New Testament connection seems to be with the people of the Old Covenant and the people of the New Covenant. The Firstborn, the people of the promise, for the most part (though obviously not all) reject salvation in Christ while many Gentiles accept it. Thus Esau would represent the Jews, the seemingly “legitimate” inheritors of the promises of God, while Jacob, the younger and at first less likable person, ends up inheriting the covenant and being the truly chosen one (“I loved Jacob but hated Esau”) and thus represents the Christians (though Esau still becomes a great nation because of his descent from Abraham, we might point out, which we could connect to the Jews still remaining an important people and heirs to the less important covenant, and we might even take as a hopeful precedent the reconciliation of Jacob and Esau later in their lives).