Paul’s Jewishness notwithstanding, it’s clear that at least by the time of 1 Corinthians, he had come to think of the Church as conceptually distinct from the Jews and from the pagans:
ἀπρόσκοποι καὶ Ἰουδαίοις γίνεσθε καὶ Ελλησιν καὶ τῇ ἐκκλησίᾳ τοῦ θεοῦ” (10:32)
Christians are, overwhelmingly, Gentile followers of the Jewish Messiah. They are neither the Chosen People set apart for the sake of being set apart (e.g., Exodus 19:6), nor pagans wallowing in idolatry. The Church for Paul has a public identity that sets apart from the considerable public demands of both Judaism and paganism: it is a society bound together by allegiance to Jesus.