Biblically speaking, being Jewish is not a race, or something that's tied to your DNA. What people claiming to be "Jewish" today assert has no bearing whatsoever on the Bible's criteria for being a Jew.
I would like to point out before we begin that claiming that any group is superior or inferior to others due to their race is the definition of racism. Many Christians, under the doctrines of Dispensationalism, have been taught that those claiming to be Jewish are superior in the sight of God, and everyone else is a filthy Gentile. This article will show that to be false.
Puzzlingly, not believing that those claiming to be Jewish are superior to everyone else will usually get you labeled as a "racist anti-semite". I expect this regardless of what I say, but I will have it known that I believe that all human beings are equal in value before God, and that no group is privy to superiority based on race - which is utterly irrelevant to God (Acts 10:34-35, 17:26, Romans 2:10-11, Isaiah 40:17, Psalm 62:9, etc.).
The way that someone became Jewish in Abraham's time was to have the males of the family circumcised (Genesis 17:10), which was a covenant that God made with Abraham in Genesis chapter 17. If someone wasn't circumcised, regardless of whether they descended from Abraham, they were to be cut off from the Jewish nation (Genesis 17:14).
Later, in the time of the Exodus, in order to become a Jew, you had to have the males circumcised and keep the Passover. If someone did this, there was to be no distinction between them and one who was "born in the land", showing that they had become a bona fide Jew and member of the nation of Israel:
Exodus 12:48
48 And when a stranger shall sojourn with thee, and will keep the passover to the Lord, let all his males be circumcised, and then let him come near and keep it; and he shall be as one that is born in the land: for no uncircumcised person shall eat thereof.
If anyone, whether born in the land or not, didn't partake in the Passover and the Feast of Unleavened Bread, they were to be cut off from the Jewish people (Exodus 12:19).
Later, when the Law of Moses was given, there was added a myriad of ways that someone could be cut off from the Jewish people, if they failed to follow certain laws (Exodus 30:33, 30:38, 31:14, Leviticus 7:20-27, 17:4-10, 18:29, 19:8, 20:3-6, 20:18, 22:3, 23:29, Numbers 9:13, 15:30, 19:13). Therefore, it's evident that being a Jew was contingent upon abiding by the Old Covenant.
We also learn, in Ezekiel 47, that a convert not born in Israel would become a member of whatever tribe they sojourned with, and therefore had an inheritance with all of the children of Israel (Ezekiel 47:22-23). This again shows that they became fully-integrated, full-fledged Jews, regardless of whether they were descended from Abraham.
If being Jewish was somehow tied to your race or DNA, it would be impossible to "become" a Jew at all, in any sense. Yet we are even given a specific example of when people "became Jews" in Esther chapter 8:
Esther 8:17
17 And in every province, and in every city, whithersoever the king's commandment and his decree came, the Jews had joy and gladness, a feast and a good day. And many of the people of the land became Jews; for the fear of the Jews fell upon them.
Since changing your race is impossible, the fact that there is provision for - and even explicit examples of - outsiders becoming Jewish, proves that being a Jew in the Old Testament had absolutely nothing to do with your birth or DNA, but rather your entrance into a covenant with the God of Abraham, Isaac, and Jacob.
In keeping with the Old Testament, the New Testament is clear that your DNA is not what makes you a Jew. A spiritual circumcision is what God takes as a token for entering the covenant, and that is predicated upon faith in Christ, not birth:
Romans 2:28-29
28 For he is not a Jew, which is one outwardly; neither is that circumcision, which is outward in the flesh:
29 But he is a Jew, which is one inwardly; and circumcision is that of the heart, in the spirit, and not in the letter; whose praise is not of men, but of God.
Philippians 3:3
3 For we are the circumcision, which worship God in the spirit, and rejoice in Christ Jesus, and have no confidence in the flesh.
Colossians 2:11
11 In whom also ye are circumcised with the circumcision made without hands, in putting off the body of the sins of the flesh by the circumcision of Christ:
From the above, we learn that anyone with the spiritual "circumcision of Christ" is a Jew, and if you lack that, God does not consider you a Jew.
Christians then, according to the above, are Jews. And, if you are not a Christian, you are not a Jew, regardless of who your parents are, according to the Bible. Read the above passages again if you need to, because I know that Zionists have told you that you are a Gentile and the "Jews" in "Israel" are Jews, but that's not what the Bible says.
We are told that God is not a respecter of persons in Acts 10:34-35 and Romans 2:10-11, and both of those statements are in context to Jews vs Gentiles. Since the New Covenant is the same for Jews and Gentiles (Galatians 3:28, Colossians 3:11), all humans are able to become a member of God's holy spiritual nation and royal priesthood (1 Peter 2:5, 9-10, Ephesians 2:11-13, 19).
God doesn't care what your DNA is, because the "flesh profiteth nothing", but rather the spirit gives life (John 6:63). That means if you were Abraham's literal child, or as the Bible says, a member of "Israel after the flesh" (1 Corinthians 10:18), it would profit you absolutely nothing unless you are spiritually right with God, which anyone who rejects Christ is not (John 3:18, 3:36).
You don't have to read 3 chapters of the New Testament before you see John the Baptist warning people about thinking that their Jewish ancestry would carry favor with God (Matthew 3:9).
With regard to assertion that modern-day "Jews" are still privy to superiority because they are the seed of Abraham, I would like to point out two key passages proving that this is a lie:
Romans 9:6-8
6 Not as though the word of God hath taken none effect. For they are not all Israel, which are of Israel:
7 Neither, because they are the seed of Abraham, are they all children: but, In Isaac shall thy seed be called.
8 That is, They which are the children of the flesh, these are not the children of God: but the children of the promise are counted for the seed.
According to the above, God says that Abraham's physical seed are not counted as his children, and in fact, only the children of the promise are counted for his seed in God's reckoning.
Who are the children of the promise? Well, the promises that were made to Abraham's seed (singular, not plural) in the Old Testament were to made to Jesus Christ:
Galatians 3:7, 16, 28-29
7 Know ye therefore that they which are of faith, the same are the children of Abraham.
16 Now to Abraham and his seed were the promises made. He saith not, And to seeds, as of many; but as of one, And to thy seed, which is Christ.
28 There is neither Jew nor Greek, there is neither bond nor free, there is neither male nor female: for ye are all one in Christ Jesus.
29 And if ye be Christ's, then are ye Abraham's seed, and heirs according to the promise.
Christians are the seed of Abraham, because Christ was the only seed of Abraham to whom the promises pertained, and we get in on that designation by virtue of Him, according to God.
Therefore, if you are a Christian, according to the above, your genealogy is:
This means that the promises made in the Old Testament to Abraham and his seed, Jesus Christ, are yours, and they do not belong to anyone who doesn't believe that Jesus is the Christ, including the modern-day fake "Nation of Israel".
Another clear passage testifying to this in found in John chapter 8:
John 8:37-39
37 I know that ye are Abraham's seed; but ye seek to kill me, because my word hath no place in you.
38 I speak that which I have seen with my Father: and ye do that which ye have seen with your father.
39 They answered and said unto him, Abraham is our father. Jesus saith unto them, If ye were Abraham's children, ye would do the works of Abraham.
Above is Jesus Christ affirming that they are literal descendants of Abraham (verse 37), and yet they are not "Abraham's children" (verse 39) because they reject Him.
Therefore, not only is it impossible for non-Christians to be "Jews", it's also impossible for them to be the "children of Abraham", according to God - their own assertions notwithstanding.
The above section is important, because if Judaism was a race based on who was descended physically from Abraham, it would constitute the vast, vast majority, if not all of, humanity.
This is because Abraham is almost certainly in everyone's family tree, which can be attested to by anyone who has actually tried to make a family tree. Every single generation, your number of ancestors doubles, and quickly balloons into quadrillions of names - as each of your ancestors has their own family tree that gets added to yours - a line of their own direct ancestors. Each of them, in turn, has their own family tree, and so on.
Obviously, these names are not all unique people. However, the odds that one person in that tree had Abraham somewhere in their family tree, is 100%. There has been so much mixing of humanity, due to traders, conquerors, migration, and so on, that no group on earth has been totally isolated with not a single outsider since the time of Abraham, until today.
This is why physical descendence from Abraham does not make one a "Jew". Such a thing is irrelevant.
The phenomenon of false Jews claiming to be Jews is not new, and was rebuked twice by the Lord:
Revelation 2:9
9 I know thy works, and tribulation, and poverty, (but thou art rich) and I know the blasphemy of them which say they are Jews, and are not, but are the synagogue of Satan.
Revelation 3:9
9 Behold, I will make them of the synagogue of Satan, which say they are Jews, and are not, but do lie; behold, I will make them to come and worship before thy feet, and to know that I have loved thee.
From the above, it's easy to see that God takes this designation very seriously. Non-Christians claiming to be Jews, when God doesn't consider them Jews, is called out explicitly, twice, as being "lies", and "blasphemy". This is not a small issue, it's one Christians should be very careful about.
Modern-day "Jews", because of their rejection of Jesus Christ, do not even worship the same God as Christians - hence the moniker "synagogue of Satan" in the verses above:
1 John 2:22-23
22 Who is a liar but he that denieth that Jesus is the Christ? He is antichrist, that denieth the Father and the Son.
23 Whosoever denieth the Son, the same hath not the Father: he that acknowledgeth the Son hath the Father also.
Again - notice that God takes this very seriously, once again calling all those who reject Jesus "liars" and "antichrists", who "hath not the Father".
Any non-Christian then, according to the Bible, does not worship God the Father, because you must have the Son to have the Father.
Going further though, Jesus Christ even made it clear that Jews do not even believe the Old Testament, otherwise they would believe in Him (John 5:45-47). The Lord says a veil is over their eyes when they try to read the Old Testament (2 Corinthians 3:14-16). So, they neither worship Jehovah, nor do they believe or understand the Old Testament, despite their claims to the contrary.
Modern-day Rabbinic Judaism is not a "race" or ethnicity, it's a religion that denies Jesus Christ, and is therefore Antichrist. They are not Jews according to God Almighty, and do not worship the same God as Christians, according to the Bible.
However, we, as Christians, are both Jews, and the children of Abraham, according to the Lord.