Why is it said that the Gospel of Matthew presents Jesus as the King of the Jews? I have heard it stated that the Gospel of Matthew presents Jesus as, specifically, the King of the Jews, rather than the Messiah, the Incarnate Word, or anything else.

What is the basis for this assertion?
The Gospel of Matthew is written for a Jewish audience, especially Palestinian Jews who were at the time oppressed by the Romans. One of Matthew's main goals in his gospel was to prove that Jesus was the true Davidic Messiah--the king the Jews were expecting, who will deliver them from oppression.

Jewish scripture foretold that the Messiah will be like a Star out of Jacob and the Scepter that rises out of Israel in Numbers 24:17. He is the One who sits on David’s throne in Isaiah 9:7. He comes with the clouds of heaven to reign over a kingdom where all people, nations, and languages, will serve Him in Daniel 7:13-14. In His kingdom the nations will no longer lift up the sword against one another in Isaiah 2:4, and His reign, we’re promised, will have no end in Isaiah 9:6-7.

And hence Jesus Christ is presented as the “Son of David, the Son of Abraham” in Matthew 1:1 and the “King of the Jews” who wise men seek to worship in Matthew 2:2. Forty times He speaks of the kingdom of heaven and says that the day would come when He would return in the clouds to establish His kingdom on earth in Matthew 26:64.

Even Jesus is presented being asked directly, “Are you the King of the Jews?” to which He replies, “I am” in Matthew 27:11. This infuriated the religious leaders who demanded He be crucified. As He hung on the cross a sign was put over His head that read, “This Is Jesus, The King Of The Jews.”