The bottom line: This wonderful question can only be clearly explained via a little history lesson.
"Israel" consisted of 12 tribes, including Judah, where the term "Jew" originated. These were the people to whom YHWH gave His Torah while they were wandering in the desert for 40 years. Although Torah has always been around since beginning of the world when YHWH commanded the universe to "be", we are told that Moshe (Moses) wrote down all the Words of YHWH (which became known as the Torah - first five Books of the Bible). YHWH also chose Moshe to present Israel (the 12 tribes) with the Ten Commandments.
It is also imperative to remember that many Egyptians (Gentiles/foreigners) followed Moshe out of Egypt (Exodus 12:38) - and they were told in Numbers 15:13-16 to do exactly as YHWH's people did once they decided to accept YHWH (the God of Abraham, Isaac and Jacob)...which means, they were to obey YHWH's Torah. Those commands were never changed! In Jeremiah 31:31-32 we see YHWH's promise to make His New Covenant NOT with the Gentiles, but with the Torah observant Houses of Israel and Judah.
The Tribe of Judah was chosen to teach the world about Torah (Genesis 49:10, Micah 4:2) and we must never forget that YHWH caused Yeshua to be born into the Tribe of Yehudah/Judah!
We learn in Romans 11 that Gentiles are "grafted in to the Olive Tree (Israel) and thus they do come into "the commonwealth of Israel" per Ephesians 2:11-16...which means they must be Torah observant (according to Numbers 15:13-16, where YHWH reiterates four times that the "foreigner" who accepts Him MUST do exactly as His "natural children" do) . However, per Zecheriah 8:23 they take hold of the tzit-tzit of a Jew and join to Judah because Y'shua is from Judah.
So, while Gentiles may not be "Jews" per se, they count as Judah - unless it is revealed to them that they were descended originally from another tribe and went away - in which case, they return to that tribe and will go through their respective gates when they enter eternity (see Revelation 21:12-13).
The Hebrew word for "to cross over" is … "ivrit" which translated into English is "HEBREW!" Avram was eventually renamed Abraham/Avraham (which is Hebrew for "High Father, Father of a multitude, many nations") after he "crossed over" thus, becoming officially known as the first "Hebrew" (Genesis 14:13).
What we're trying to say is that Gentiles ... who were considered "pagan" before they "accepted Jesus" (Yahweh's Divine Son whose Hebrew name was Yeshua - who was the Word, John 1:1, 14; and the "arm of Yahweh" - Isaiah 53:1) actually become "proselytes" - "righteous Gentiles" … and more to the point: "HEBREWS" (wich means that men must become circumcised (Genesis 17: 7-14, Exodus 12:48-51, 1 Corinthians 7:18-19), which ultimately renders them indistinguishable from the native born Jew. As we are told in Romans 12:5, we are ONE in Messiah!