The answer to the question will not be complete without discussing whom Jesus considered as his brothers. See Matt 25:37-40 (KJV):

³⁷ Then shall the righteous answer him, saying, Lord, when saw we thee an hungred, and fed thee? or thirsty, and gave thee drink? ³⁸ When saw we thee a stranger, and took thee in? or naked, and clothed thee? ³⁹ Or when saw we thee sick, or in prison, and came unto thee? ⁴⁰ And the King shall answer and say unto them, Verily I say unto you, Inasmuch as ye have done it unto one of the least of these my brethren, ye have done it unto me.

Did the Lord mean to restrict the field of Christian charity and compassion only to his blood relatives? By no means!

In fact, the terms denoting 'brother' including the counterparts of the Greek word adelphoí, lit. "of the same womb", are still used in oriental countries eg. India to encompass a wider set of relationships including cousins. To refer to siblings born out of the same set of parents they use terms like 'own brother'.

For whosoever shall do the will of my Father which is in heaven, the same is my brother, and sister, and mother. Matthew 12:50 KJV.

In this first verse, μου αδελφος και αδελφη και μητηρ εστιν means what it says : brother and sister and mother.

After this I beheld, and, lo, a great multitude, which no man could number, of all nations, and kindreds, and people, and tongues, stood before the throne, and before the Lamb, clothed with white robes, and palms in their hands; Revelation 7:9 KJV

In this second verse it is clear from many other concomitant scriptures, but especially those within the Apocalypse, that this vast and (by man) innumerable company are the same as Jesus describes in the first verse, namely, those who do the will of the Father of Jesus Christ.

Therefore, in the New Testament, that is to say within the compass of all that is called 'New Testament' the number of such relatives of Jesus Christ cannot be numbered by human means.

The New Testament is not about human relationships. It is about relationships in Spirit ; it is about a birth from above ; it is about another humanity than that of Adam ; it is about the Father and the Son.

I looked at "Bibelwissenschaft" finding this (translated to english by deepl.com)

(...) Mark 6:1-6 tells of the rejection of Jesus in → Nazareth, with those present at his teaching in the synagogue referring to Jesus as "son of Mary" and mentioning four brothers of Jesus named James, Joses, Judas and Simon, as well as sisters of Jesus in the plural. From this text, we can conclude that Mary had at least seven children. However, later interpretations of the text have understood the siblings of Jesus - based on the idea of Mary's perpetual virginity - as half-siblings of Jesus (children of Joseph from a previous marriage) or as cousins of Jesus. This is a dogmatic reading, advocated mainly by the Catholic side, but it is not close in the context of the New Testament. In the writings of the New Testament, the brothers (and sisters) of Jesus are also spoken of unabashedly in other places (cf. Mt 12:46-50; Mt 13:55-56; Mk 3:31-35; Lk 8:19-21; Jn 2:12; Jn 7:3-10; Acts 1:14; 1Cor 9:5); the brother of Jesus, James, plays a special role (cf. Acts 15:13; 1Cor 15:7; Gal 1:19; Gal 2:9).
(...)
Translated with www.DeepL.com/Translator (free version)

This goes on a bit further, but I thought this is the essential part for your question, and gives a compact list of references. Hope this helps...

To add some background about the "mother and brothers" passage from Lesly's post.

OP: Does anybody actually claim to be a brother or sister by virtue of being children of either Joseph or Mary?

Christ makes the claim.

While he [Christ] yet talked to the people, behold, his mother and his brethren stood without, desiring to speak with him. Mt 12:46

So, who are His mother and brothers? Is it Mary? Yes, no doubt. What about His brothers? Are they cousins? Are they sons of Joseph from his prior marriage? Or are they children of Mary and Joseph, His brothers?

Then one said unto him, Behold, thy mother and thy brethren stand without, desiring to speak with thee. Mt 12:47

The townsfolk knew them, His mother and brothers. But who?

But he answered and said unto him that told him, Who is my mother? and who are my brethren? Mt. 12:48

The point is they knew He had a mother and brothers. Here's Tertullian.

And therefore, when to the previous question, “Who is my mother, and who are my brethren?”4210 He added the answer “None but they who hear my words and do them,” He transferred the names of blood-relationship to others, whom He judged to be more closely related to Him by reason of their faith. Now no one transfers a thing except from him who possesses that which is transferred. If, therefore, He made them “His mother and His brethren” who were not so, how could He deny them these relationships who really had them? Surely only on the condition of their deserts, and not by any disavowal of His near relatives; teaching them by His own actual example,4211 that “whosoever preferred father or mother or brethren to the Word of God, was not a disciple worthy of Him.”4212 Besides,4213 His admission of His mother and His brethren was the more express, from the fact of His unwillingness to acknowledge them. That He adopted others only confirmed those in their relationship to Him whom He refused because of their offence, and for whom He substituted the others, not as being truer relatives, but worthier ones. Finally, it was no great matter if He did prefer to kindred (that) faith which it4214 did not possess.4215 AM IV XIV

At that time, the early church was keen on making sure that the argument against Marcion and Valentinus, who argued Christ had no flesh or if flesh only appeared as such, was solid and sure.

But whenever a dispute arises about the nativity, all who reject it as creating a presumption in favour of the reality of Christ’s flesh, wilfully deny that God Himself was born, on the ground that He asked, “Who is my mother, and who are my brethren?”7038 Let, therefore, Apelles hear what was our answer to Marcion in that little work, in which we challenged his own (favourite) gospel to the proof, even that the material circumstances of that remark (of the Lord’s) should be considered.7039 First of all, nobody would have told Him that His mother and brethren were standing outside, if he were not certain both that He had a mother and brethren, and that they were the very persons whom he was then announcing,—who had either been known to him before, or were then and there discovered by him; although heretics7040 have removed this passage from the gospel, because those who were admiring His doctrine said that His 528 supposed father, Joseph the carpenter, and His mother Mary, and His brethren, and His sisters, were very well known to them. On the Flesh of Christ VII

So, by showing up, His mother and brothers were making the claim to be exactly that. Christ, however, points to those who share His faith as His true relations.

How many different people in the New Testament claim to be "Brothers" or "Sisters" of Jesus?

To my way of seeing how this question is worded, the only answer could be no one personally ever claimed to be a "Brother" or "Sister” of Jesus? in the New Testament.

That said, we see in Matthew 13:54-57 the crowd making the claim that James, Joseph, Simon and Jude were his brothers and he had several sisters. But these individuals never personally claimed to be Jesus’ brother or sister!

54 Coming to his hometown, he began teaching the people in their synagogue, and they were amazed. “Where did this man get this wisdom and these miraculous powers?” they asked. 55 “Isn’t this the carpenter’s son? Isn’t his mother’s name Mary, and aren’t his brothers James, Joseph, Simon and Judas? 56 Aren’t all his sisters with us? Where then did this man get all these things?” 57 And they took offense at him.

But Jesus said to them, “A prophet is not without honor except in his own town and in his own home.”

The brothers and/or sisters of Jesus are mentioned several other times in the Scriptures but again no individual ever claimed to be personally his brother or sister!

• John 7:1-10

• Mark 6:2-3

• Galatians 1:18-20

• Mark 3:34–35: Jesus declared that those who follow Him are His brothers: “He looked at those seated in a circle around him and said, ‘Here are my mother and my brothers! Whoever does God’s will is my brother and sister and mother.’”

Some have theorized that the Greek words adelphos (“brothers”) and adelphai (“sisters”) in Matthew 13 refer to spiritual brothers and sisters. Others, who hold to the idea of Mary’s perpetual virginity, assume that the references to Jesus’ siblings simply mean that Joseph had children of his own, before his marriage to Mary. However, there is no biblical support for either theory, and we have no logical reason to believe that the siblings mentioned by name in Scripture were not the biological children of both Mary and Joseph. We are never told exactly how many siblings Jesus had, but Mark 6:3 indicates there had to have been at least six: at least four brothers, named; and at least two sisters, unnamed. - How many siblings did Jesus have?

The Pharisees themselves have serious doubts about Jesus’ past and can not understand him fully nor the true genealogy of Christ!

We know that God spoke to Moses, but as for this fellow, we don’t even know where he comes from. - John 9:28.

There is a far fetched story in the Apocryphal readings of the Gospel of Thomas that the Apostle Thomas Didymos was the twin brother of Jesus!

This book opens with the lines, "These are the secret words which the living Jesus spoke, and the twin, Didymos Judas Thomas wrote them down." Then there follows a list of the sayings of Jesus. Now this raises all kinds of questions. Did Jesus have a twin brother? Actually the name Thomas Didymos - well, Thomas is Hebrew for twin. Didymos is Greek for twin.... The implication here is that he is Jesus' twin. But this character, of course, also appears in the Gospel of John, he's one of the disciples, the twin. Here he appears as if he's Jesus' twin, and he is one who knows secret teaching, which Jesus hasn't given to all other people. Some of these sayings are familiar. We know them from Matthew and Luke - Jesus said, "I have come to cast fire on the earth." Or "Behold, a sower went out to sow," and so forth.... Others are as strange and compelling as Zen koans. My favorite of these is saying number 70, which says, "If you bring forth what is within you, what you bring forth will save you. If you do not bring forth what is within you, what you do not bring forth will destroy you." The gospel opens as Jesus invites people to see....

Strange as this last one may seem, some actually may believe it: Bible mystery: Was the aspotle Thomas actually Jesus Christ's twin brother?

Once again I would like to state that no one personally and clearly makes the claim that they were Jesus’ brother or sister.

The following may be of interest to some:

• The Brethren of the Lord

• How Many Brothers and Sisters Did Jesus Have?

TLDR: No one seems to specifically claim to be a blood sibling of Jesus in the New Testament. However, 4 siblings are named by other people in the Gospels, and two are possibly corroborated by the Epistles

I looked through every instance of the word "brother" or "sister in the NIV New Testament.

The Gospels

There are four references to Jesus' siblings in the Gospels:

Matthew 12:46-50 (and Mark 3:31-35, and Luke 8:19-21)

While Jesus was still talking to the crowd, his mother and brothers stood outside, wanting to speak to him. Someone told him, “Your mother and brothers are standing outside, wanting to speak to you.”

He replied to him, “Who is my mother, and who are my brothers?” Pointing to his disciples, he said, “Here are my mother and my brothers. For whoever does the will of my Father in heaven is my brother and sister and mother.”

Matthew 13:55-56 (and Mark 6:2-3)

Isn’t this the carpenter’s son? Isn’t his mother’s name Mary, and aren’t his brothers James, Joseph, Simon and Judas? Aren’t all his sisters with us? Where then did this man get all these things?”

John 2:12

After this he went down to Capernaum with his mother and brothers and his disciples. There they stayed for a few days.

and John 7:1-11 (showing v3-5)

Jesus’ brothers said to him, “Leave Galilee and go to Judea, so that your disciples there may see the works you do. No one who wants to become a public figure acts in secret. Since you are doing these things, show yourself to the world.” For even his own brothers did not believe in him.

So in the Gospels, we see that Jesus has siblings, both brothers and sisters, and four of his brothers are named: James, Joseph, Simon and Judas.

The Rest of the New Testament

In the rest of the New Testament, things get a bit blurry, since sometimes believers in general are named brothers or sisters of Christ. But there are a few places that seem to refer to people as blood relatives of Jesus.

Acts 1:14

They [the apostles] all joined together constantly in prayer, along with the women and Mary the mother of Jesus, and with his brothers.

1 Corinthians 9:5

Don’t we have the right to take a believing wife along with us, as do the other apostles and the Lord’s brothers and Cephas?

The only place in the New Testament outside of the Gospels that seems to explicitly name a blood brother of Christ is Galatians 1:18-20. James (not the apostle by context) is named by Paul as "the Lord's brother"

Then after three years, I went up to Jerusalem to get acquainted with Cephas and stayed with him fifteen days. I saw none of the other apostles—only James, the Lord’s brother. I assure you before God that what I am writing you is no lie.

Finally, people speculate that Jude "a brother of James" is the same as the Judas named in Matthew 13, and that the James he names as his brother is the same James named there and in Galatians 1

So, no one seems to specifically claim to be a blood sibling of Jesus in the New Testament. However, 4 siblings are named by other people in the Gospels, and two are possibly corroborated by the Epistles

Matthew 13:35-36 answers:

Is not this the carpenter’s son? Is not his mother called Mary? And are not his brothers James and Joseph and Simon and Judas? And are not all his sisters with us?

The sisters are not named, so we cannot say how many (half) brothers and sisters Jesus had in total.

In John 7:1-10, His brothers go on to the festival while Jesus stays behind. In Acts 1:14, His brothers and mother are described as praying with the disciples. Galatians 1:19 mentions that James was Jesus’ brother.

“But he (Joseph) had no union with her (Mary) until she gave birth to a son. And he gave Him the name Jesus” (Matthew 1:25).

Protestants conclude that these passages mean Jesus had actual blood half-siblings. Four sons plus some sisters born to Mary after she gave birth to Jesus.