There are many kinds of brotherhoods (or sisterhoods). There is the universal brotherhood of all mankind. There are national brotherhoods of fellow citizens who embrace the same national identity, language, and culture. There are fraternal organizations wherein members call each other brother such as the International Brotherhood of Teamsters and the Fraternal Order of Police. Of course, there is the literal brotherhood of siblings and even relatives. Then there is the highest brotherhood of all - the spiritual one. It is the highest because it is spiritual in nature, and because it will last into eternity. All other fraternities are based in this life only and will not translate into eternity.
In the Old Testament, the word brother is broader than just literal siblings. It can mean male relative. The first example of this is Abraham's nephew Lot (Gen 14:14). In the New Testament this is not the case. Some try to say that Jesus did not have literal brothers and sisters as described in Matthew 13:55-56, but He did. The first brothers were Cain and Abel, but Cain hated and murdered Abel (Gen 4). Isaac and Ishmael were half-brothers, but the descendants of Isaac became the Israelites, and the descendants of Ishmael became the Arabs (Gen 16-17 & 22). Jacob and Esau were twin brothers, but they too did not get along (Gen 27-33). Jacob's descendants became the nation of Israel, and Esau's became the Edomites. Esau's hatred of Jacob lived on in his descendants for hundreds of years (Num 20, Ps 137:7, Eze 25, Joel 3:19, Amos 1:11). Joseph's brothers turned against him (Gen 37). There was to be a brotherhood of Israelites (Ex 4:18, Lev 19:17, Lev 25:25-39 & 46, Num 32:6, Deut 15, Deut 17:15, Deut 19:18-19, Deut 22:1-2, Deut 23:19-20, Deu 24:7-14, Deut 25:3).
The greatest brotherhood that has ever existed is the brotherhood of Christians. That is because it is based on the greatest covenant in the history of man - the new testament made by the blood of Jesus Christ (Mt 26:28). God's covenant with Israel was great, but it was fulfilled by Jesus Christ, although there are still promises to be fulfilled for Israel in the future. This is the main topic of the book of Hebrews. Throughout the New Testament, fellow believers in Jesus are called brothers.
It is natural and normal to desire to be part of a group. Except for extreme and unusual cases, most people want to belong. We want to be accepted and validated by our peers. We need a support network of people around us. As John Donne said, "No man is an island". That is one of the many ways the church is valuable. It provides a community of people of like faith to be a part of. Christians are to cherish this relationship and participate in it. "Honour all men. Love the brotherhood. Fear God. Honour the king (1Pet 2:17)." John put it in the strongest terms in 1John 2-4. It can be summarized in the admonition to love your brother.
How does one join this brotherhood? We do not say a secret oath. We do not sign a membership card. We do not take a pledge. We do not pay a membership fee. We are born again by repenting of our sins, being baptized by immersion in water in the name of Jesus Christ for the remission of sins and receiving the gift of the Holy Ghost with the initial outward evidence of speaking in an unknown language (Jn 3:1-7, Acts 2:1-4 & 38-39, Acts 8:12-17, Acts 10:43-48, Acts 19-17). We join the brotherhood by becoming a child of God (Jn 1:12, Rom 8:14, Phil 2:15, 1Jn 3:1-2). Then, other children of God become our brothers and sisters.
In the story of the rich man and Lazarus, the rich man had brothers who had not died yet (Lk 16:19-31). He asked Abraham to send Lazarus back to life on this earth to testify to his relatives so they would not end up in hell. Even in hell the rich man understood the value of brotherhood. Also, he called Abraham father, and Abraham called him son, although he was in hell. He was still Jewish although he had not abided in the brotherly covenant of Moses and lost his soul. So too, Christians can become children of God and still be lost. That will be an additional torment for those people. They were saved and had become children of God. They were on their way to eternal life but had forfeited the benefits of their sonship through sin.
Christians need to guard the brotherhood against attempts to delude and dilute its value and meaning. There have always been efforts to do so, there are now, and they will continue until the end. There were attempts to dilute in in the Old Testament. For example, in the days of Ezra and Nehemiah, there were people who attempted to identify with the brotherhood who had not entered into the accompanying covenant, and they were rejected by the leaders (Ezra 4-6 & 9-10, Neh 4 & 13). They were very careful to specifically detail who belonged to it and who did not, and they documented it by genealogy (Ezra 2, 8, and 10; Neh 3, 7, 10-12). Today these efforts include diluting Biblical doctrine and how to receive salvation so that people who have not followed the apostles' doctrine and been born again are considered part of the brotherhood. This is called ecumenicism. It is a lying spirit. It sounds good on the surface. It sounds like it is all about love, acceptance, and unity, but it is nothing but deceit. Biblical unity can only be based on scriptural truth. Biblical love can only be based on scriptural truth. Anything that is placed above truth is corrupted. When Christians prioritize "love" and "unity" over truth, both are corrupt versions of them. Christians can love people without diluting the brotherhood. How is it love to tell someone they are saved brothers when they have not obeyed the gospel? That is a damnable heresy that can condemn people's souls for eternity, which is the opposite of the alleged purpose of being "inclusive". Telling them the truth in love with wisdom is the thing to do, not telling them love without truth. Telling them how to join the brotherhood rather than telling them they already have before they do is what Christians ought to be doing.
Before Paul became a Christian, he persecuted the church (Acts 8). God confronted him while he was on the way to Damascus to persecute Christians there, and he repented. The Lord sent a Christian named Ananias to Paul to tell him how to be saved (Acts 9). Pau was baptized and received the Holy Ghost. Ananias called Paul brother before he was baptized. Ananias and Paul were both Jews as were all Christians at that point (Acts 10-11). They were also fellow humans. For confirmation of this meaning, let us look and Paul's own writings. Paul affirmed the brotherhood of all men as the creation of God on Mars Hill in Athens to the Greeks (Acts 17:24-31). He also spoke of his brotherhood with his fellow Jews (Rom 9:3-4). However, he taught clearly and repeatedly that although the church certainly welcomes all who sincerely seek to be Christians and obey the truth, and even should be involved to some degree with those without the brotherhood (Col 4:5, 1Thes 4:12, 1Tim 3:7), there is a clear distinction between believers and non-believers. The fact that these three verses use the phrase "them that are without" shows this, besides many other verses in his writings. He never called non-Christians brothers in the faith, and made distinction between them (1Cor 5:11, 1Cor 6:6, etc.).
Another way we can see a clear distinction between those who should be called brothers and those who should not is in the use of the word "world". This word has two meanings. The first is the people in the world, as in "God so loved the world, that He gave his only begotten Son...(Jn 3:16)". God gave His Son so all could be saved if they believe in Jesus the way the Bible says to. The other meaning is the value system that the majority of people live by (1Jn 2:15-17). Those who follow that way are excluded from the brotherhood (1Cor 6:9-11, Jam 4:1-4).
Christians are given the ministry and word of reconciliation (2Cor 5:18-20). They are ambassadors for Christ. They are to invite people to reconciliation and citizenship in the kingdom of God (Phil 3:20-21). The word conversation here means citizenship. An ambassador does not give people citizenship in their home nation just because they ask. The person desiring citizenship has to formally apply and go through a process. People desiring citizenship in the kingdom of God and sonship in God are to be told how to receive that and should not be told they have done so before they do. Once they do, they have access to the greatest fellowship and citizenship in all eternity.