Why is it said that the Gospel of Matthew presents Jesus as the King of the Jews?

Perhaps Matthew has a theme of Jesus as King of the Jews, but certainly no more so than, for example, John's Gospel. While there are only about 5 actual references to Jesus as King of the Jews in Matthew, I also count 5 in Mark and 3 in Luke, with at least 11 such references in John, including the dramatic account of Pontius Pilate insisting on this being placed on the sign at the crucifixion:

John 19:19-22: And Pilate wrote a title, and put it on the cross. And the writing was, JESUS OF NAZARETH THE KING OF THE JEWS. This title then read many of the Jews: for the place where Jesus was crucified was nigh to the city: and it was written in Hebrew, and Greek, and Latin. Then said the chief priests of the Jews to Pilate, Write not, The King of the Jews; but that he said, I am King of the Jews. Pilate answered, What I have written I have written.

Matthew's genealogy shows Joseph as being descended from King David, but so also does Luke's slightly variant genealogy, with neither portraying Jesus as king because of his descent. The wise men from the east, coming to look for the new-born king of the Jews, provide the strongest support for seeing Matthew as defining Jesus as King of the Jews.

More than anything, Matthew seems to portray Jesus as Son of God beginning with his conception by the Holy Spirit:

Matthew 1:18: Now the birth of Jesus Christ was on this wise: When as his mother Mary was espoused to Joseph, before they came together, she was found with child of the Holy Ghost.

Matthew proclaims Jesus as King

There are many contextual clues within Matthew geared towards a Jewish audience, hinting continually at Jesus being 'Messiah', or the true anointed King of Israel. As modern western Christians, we can easily miss and mis-read many of these. For us a Christ/Messiah just means a spiritual saviour, and so until we read all clues to Messiah-ship as translating as 'King' to a Jew, we don't quite see Matthew's words in their proper light.

A King from the Beginning

"The book of the genealogy of Jesus Christ, the son of David, the son of Abraham." - Matthew 1:1, cf. v20

Matthew sets Jesus as 'the son of David', or the true heir to the promise of the Davidic kings as set forth in 2 Samuel 7, that there would always be a king to sit on David's throne. Reading the genealogy that follows, Matthew chooses a different line from Luke, carefully tracing Jesus down the proper line of Davidic kings who actually sat on the throne. He arranges the members of the line in groups of seven, the perfect number, as well as the number of David's name (דָּוִד).

It is good to note that 'Son of God' is also a very common term for Jesus throughout Matthew, but this does link back through the Davidic promises too (cf. 2 Samuel 7:14, Psalm 2:7).

A King in the Middle

For a start, Matthew uses the title Christ no less than Seventeen times (concentrated at beginning and end of the book), compared with twelve occurrences in Luke and eight in Mark. John as a non-synoptic does win with nineteen references, however. But Matthew makes this theme very clear:

A few highlights:

• Chapter Two wise men from the east come identifying the child as 'the King of the Jews'.
• Chapter Three has echoes of 1 Samuel 10/16, where the new king is blessed by the Levite prophet, and the Spirit of the Lord rushes upon him to confirm his calling (but I'll admit that one is more of a stretch).
• Chapter Four and thereafter our well-identified King goes proclaiming a 'Kingdom' is at hand (4:17,23).
• Once the sermon on the mount concludes, Matthew resumes his proofs of Jesus' messiah-ship by fulfilling various messianic prophecies (8:17)
• Chapter Nine has the two men come out calling Jesus "Son of David (9:27)
• In 9:36 Jesus begins to refer to the people as 'sheep without a shepherd'. This phrase goes back to the anointing of Joshua (Yeshua!) in Num 27:17, but similar references repeat throughout the Old Testament and culminate in Ezekiel's prophecy:

"And I will set up over them one shepherd, my servant David, and he shall feed them: he shall feed them and be their shepherd. And I, the Lord, will be their God, and my servant David shall be prince among them. I am the Lord; I have spoken." - Ezekiel 34:23-24

• In Chapter Ten he gives the disciples 'Authority', and sends them out to preach the 'Kingdom' bringing in lost sheep (10:6)
• In Chapter Twelve upon one of his healings, the crowd declare "Could this perhaps be the Son of David?" (12:22-23)
• In Chapter Fifteen a woman comes out calling upon Jesus, "Have pity on me, Lord, Son of David!" (15:22)
• In Chapter Sixteen we get the firmest yet confession "You are the Christ, the Son of the living God!" (16:15), and from here his sights are set on Jerusalem, the "city of the great King" (Mt 5:35)
• In Chapter Twenty two blind men ask for mercy from Jesus, the "Son of David" (20:29)

A King at the End

• In Chapter Twenty-One Jesus is entering Jerusalem, and does so in clear kingly style: "Say to the daughter of Zion, 'Behold! Your king comes to you gentle and mounted on a donkey" (21:5) and the people reinforce it, "Hosanna to the Son of David!" (21:9)
• In Chapter Twenty-One Jesus is teaching in the temple, and the children shout out "Hosanna to the Son of David!" (21:15)
• In Chapter Twenty-Two Jesus' teaches about the 'son of a king' giving a wedding feast in reference to himself (22:21)
• In Chapter Twenty-Four Jesus threatens the destruction and rebuilding of the temple, which is the proper role of the 'Son of David' given in 2 Samuel 7, for David is not authorised to tear down the tent of God and build a temple, but the Son of David is.
• In Chapter Twenty-Six he is quite literally 'anointed' with oil, which is the sign done over new kings/messiahs throughout scripture.
• In Twenty-Seven Pilate's first question is "Are you King of the Jews?" (27:11) and Jesus affirms it. Again v21, "Jesus who is called Christ"
• In Twenty-Seven the soldiers make Jesus a mock crown and robe, and mock him as "King of the Jews!" (27:28)
• In Twenty-Seven there is a sign above his head which simply says "This is Jesus, King of the Jews" (27:37)

Conclusion

Some of the points above are disputable, but on the whole, especially considering the Jewish connotations of a 'Messiah/Christ' being an anointed King, it is inescapable to see a theme of "Jesus as King" in the gospel of Matthew. This is throughout the text beginning to end, both stated and implied. He does not appear to bring most of these claims from Jesus himself, but rather tends to quote others and explain situations in such a way as to emphasise the kingly credentials and nature of Jesus.