What commentary or traditions, if any, exist concerning the identity of the person named Joseph in Matthew 13?

James the Just (son of Joseph, brother of the Lord) James the Less (son of Alphaeus, apostle) James the Greater (son of Zebedee, apostle)

It is highly unlikely that the mentioned James in Mt 13:55 is anyone else than James the Just, first bishop of the Jerusalem church, the Lord's brother (same mother, different father). Josephus, Ignatius, and many others mention this.

Festus was now dead, and Albinus was but upon the road; so he assembled the sanhedrim of judges, and brought before them the brother of Jesus, who was called Christ, whose name was James, and some others, [or, some of his companions]; and when he had formed an accusation against them as breakers of the law, he delivered them to be stoned http://www.biblestudytools.com/history/flavius-josephus/antiquities-jews/book-20/chapter-9.html

If thou wilt give me leave, I desire to go up to Jerusalem, and see the faithful saints who are there, especially Mary the mother, whom they report to be an object of admiration and of affection to all. For who would not rejoice to behold and to address her who bore the true God from her own womb, provided he is a friend of our faith and religion? And in like manner [I desire to see] the venerable James, who is surnamed Just, whom they relate to be very like Christ Jesus in appearance, in life, and in method of conduct, as if he were a twin-brother of the same womb. They say that, if I see him, I see also Jesus Himself, as to all the features and aspect of His body. Moreover, [I desire to see] the other saints, both male and female. Alas! why do I delay? Why am I kept back? Kind teacher, bid me hasten [to fulfil my wish], and fare thou well. Amen. ANF01. The Apostolic Fathers with Justin Martyr and Irenaeus | Christian Classics Ethereal Library

Origen says this.

… these disasters happened to the Jews as a punishment for the death of James the Just, who was a brother of Jesus (called Christ),—the Jews having put him to death, although he was a man most distinguished for his justice.Paul, a genuine disciple of Jesus, says that he regarded this James as a brother of the Lord, not so much on account of their relationship by blood, or of their being brought up together, as because of his virtue and doctrine. http://www.ccel.org/ccel/schaff/anf04.vi.ix.i.xlviii.html

So, early tradition identifies James the Just as the brother mentioned in Mt. 13:33. From Jamieson Fausset Brown, we find this commentary on the brothers.

The confusion about who this James was arose because of Jerome who thought Joseph should also be ever-virgin, like he thought, was Mary.

As to the names here mentioned, the first of them, "JAMES," is afterwards called "the Lord's brother" (see on JF & B for Ga 1:19), but is perhaps not to be confounded with "James the son of Alphaeus," one of the Twelve, though many think their identity beyond dispute. This question also is one of considerable difficulty, and not without importance; since the James who occupies so prominent a place in the Church of Jerusalem, in the latter part of the Acts, was apparently the apostle [but only after resurrection, while James the Less was an apostle prior (and after)], but is by many regarded as "the Lord's brother," while others think their identity best suits all the statements. The second of those here named, "JOSES" (or Joseph), must not be confounded with "Joseph called Barsabas, who was surnamed Justus" ( Act 1:23 ); and the third here named, "SIMON," is not to be confounded with Simon the Kananite or Zealot (see on JF & B for Mt 10:4). These three are nowhere else mentioned in the New Testament. The fourth and last-named, "JUDAS," can hardly be identical with the apostle of that name--though the brothers of both were of the name of "James"--nor (unless the two be identical, was this Judas) with the author of the catholic Epistle so called. -source-

So, tradition would say that James the Just the Lord's brother like the others did not believe and so was different from the two James who were apostles prior to (and after) resurrection.

Subsequent, we learn this.

There still survived of the kindred of the Lord the grandsons of Judas, who according to the flesh was called his brother. These were informed against, as belonging to the family of David, and Evocatus brought them before Domitian Caesar: for that emperor dreaded the advent of Christ, as Herod had done. So he asked them whether they were of the family of David; and they confessed they were. https://ccel.org/ccel/schaff/npnf201/npnf201.iii.viii.xx.html#fnf_iii.viii.xx-p2.2

Cyril of Jerusalem circa 350 CE helps to clarify all of this.

And He was seen of Cephas, then of the twelve; (for if thou believe not the one witness, thou hast twelve witnesses then He was seen of above five hundred brethren at once ; (if they disbelieve the twelve, let them admit the five hundred after that He was seen of James, His own brother, and first Bishop of this diocese. Seeing then that such a Bishop originally saw Christ Jesus when risen, do not thou, his disciple, disbelieve him. But thou sayest that His brother James was a partial witness; afterwards He was seen also of me Paul, His enemy; and what testimony is doubted, when an enemy proclaims it? NPNF2-07. Cyril of Jerusalem, Gregory Nazianzen | Christian Classics Ethereal Library

James His own brother. Can't get any clearer.

Finally, Eusebius relates this, distinguishing between James the Just (Lord's brother) and James the Lessor (son of Zebedee).

4. But the same writer, in the seventh book of the same work, relates also the following things concerning him: “The Lord after his resurrection imparted knowledge to James the Just and to John and Peter, and they imparted it to the rest of the apostles, and the rest of the apostles to the seventy, of whom Barnabas was one.245 But there were two Jameses:246 one called the Just, who was thrown from the pinnacle of the temple and was beaten to death with a club by a fuller,247 and another who was beheaded.”248 Paul also makes mention of the same James the Just, where he writes, “Other of the apostles saw I none, save James the Lord’s brother.” -source-

So, to answer the OP.

The apostles, prior to resurrection, are named at Mt 10:2-4, Mk 3:16-19, Lk 6:14-16, Acts 1:13.

James the Just son of Joseph is different from James the Less son of Zebedee (which incidentally, if one confuses the two, then we lose the meaning of renaming sons of Zebedee to sons of thunder).

As to naming children the same, I'm not sure, but here and here may help.

Regarding Joseph/Joses, not much is known past his mention in the Bible as quoted above. However, there is the tradition he may have followed in the footsteps of his brother James the Just.

A "Joses" appears in the bishop lists of Epiphanius ("Josis") and Eusebius ("Joseph") of the early bishops of Jerusalem.[19] -source-

So, if not mistaken, all OP questions are now answered.

What commentary or traditions, if any, exist concerning the identity of the person named Joseph in Matthew 13?

The Gospel of Matthew was probably written in Aramaic. The Aramaic word for brother can refer to brother, half-brother, step-brother or adoptive brother!

Origen points to a tradition that James, Joseph, Simon and Judas were the sons of St. Joseph through a previous marriage.

The Brethren of Jesus

And they spoke, wondering, (not knowing that He was the son of a virgin, or not believing it even if it was told to them, but supposing that He was the son of Joseph the carpenter,) is not this the carpenter's son? Matthew 13:55 And depreciating the whole of what appeared to be His nearest kindred, they said, Is not His mother called Mary? And His brethren, James and Joseph and Simon and Judas? And His sisters, are they not all with us? Matthew 13:55-56 They thought, then, that He was the son of Joseph and Mary. But some say, basing it on a tradition in the Gospel according to Peter, as it is entitled, or The Book of James, that the brethren of Jesus were sons of Joseph by a former wife, whom he married before Mary. Now those who say so wish to preserve the honour of Mary in virginity to the end, so that that body of hers which was appointed to minister to the Word which said, The Holy Ghost shall come upon you, and the power of the Most High shall overshadow you, Luke 1:35 might not know intercourse with a man after that the Holy Ghost came into her and the power from on high overshadowed her. And I think it in harmony with reason that Jesus was the first-fruit among men of the purity which consists in chastity, and Mary among women; for it were not pious to ascribe to any other than to her the first-fruit of virginity. And James is he whom Paul says in the Epistle to the Galatians that he saw, But other of the Apostles saw I none, save James the Lord's brother. Galatians 1:19 And to so great a reputation among the people for righteousness did this James rise, that Flavius Josephus, who wrote the Antiquities of the Jews in twenty books, when wishing to exhibit the cause why the people suffered so great misfortunes that even the temple was razed to the ground, said, that these things happened to them in accordance with the wrath of God in consequence of the things which they had dared to do against James the brother of Jesus who is called Christ. And the wonderful thing is, that, though he did not accept Jesus as Christ, he yet gave testimony that the righteousness of James was so great; and he says that the people thought that they had suffered these things because of James. And Jude, who wrote a letter of few lines, it is true, but filled with the healthful words of heavenly grace, said in the preface, Jude, the servant of Jesus Christ and the brother of James. Jude 1 With regard to Joseph and Simon we have nothing to tell; but the saying, And His sisters are they not all with us, Matthew 13:56 seems to me to signify something of this nature — they mind our things, not those of Jesus, and have no unusual portion of surpassing wisdom as Jesus has. And perhaps by these things is indicated a new doubt concerning Him, that Jesus was not a man but something diviner, inasmuch as He was, as they supposed, the son of Joseph and Mary, and the brother of four, and of the others — the women— as well, and yet had nothing like to any one of His kindred, and had not from education and teaching come to such a height of wisdom and power. For they also say elsewhere, How knows this man letters having never learned? John 7:15 which is similar to what is here said. Only, though they say these things and are so perplexed and astonished, they did not believe, but were offended in Him; as if they had been mastered in the eyes of their mind by the powers which, in the time of the passion, He was about to lead in triumph on the cross.

Commentary on the Gospel of Matthew (Book X)

The Jewish tradition of not naming one's son after his father seems not to have existed at the time in question. For example a study of 2nd century CE Jewish business documents in Scripta Classica Israelica vol. XX 2001 pp. 169-178 has provided at least three instances of sons being named after their fathers.

• Simon son of Simon
• Judah son of Judah
• Jesus son of Jesus

Since these documents represent a relatively small sample, it is logical that there were many such examples. There is no known evidence of a tradition that exists in some other cultures, where the first son is named after his father. Thus, such a son's place in the family would be a matter of conjecture or theological opinion. Regarding denominational traditions, Catholic commentators consider this Joseph/Joses to be either a cousin of Jesus or a son of Joseph by a previous marriage. Some Protestant commentators take the some viewpoint but others have no objection to his being a younger brother of Jesus. Gill's commentary covers all the bases when it says:

his brethren; not strictly so, but either the sons of Joseph by a former wife; or Mary's, or Joseph's brothers or sisters sons, and so cousins to Christ; it being usual with the Jews to call such, and even more distant relations, brethren

From the above we can deduce that it is certainly possible that Joseph (the supposed father of Jesus) would have named one of his sons "Joseph." Beyond that, there are many opinions.