Matthew 3:16 - 17 describes Jesus baptism -- not his anointing as a king. Baptizing does not equal messianic anointing. The fact that Matthew also says Jesus "saw" the holy spirit descending on him is meaningless --
Since there is no "holy spirit" (ghost) what did the prophet Isaiah mean in Y'shayahu / Isaiah 11:2 when he said that "And the spirit of the L-rd shall rest upon him, a spirit of wisdom and understanding, a spirit of counsel and heroism, a spirit of knowledge and fear of the L-rd"?
Perhaps it is time to refresh the levels of "connectedness" humans have with G-d. The highest form of connection is prophecy -- which is direct communication from G-d to a human being. Moses conversed with G-d, but all other prophets communicated with Him through visions and dreams. The role of a prophet is to relay his or her message from G-d to his / her own generation.
Divine inspiration, or a spirit of holiness, is a level below prophecy. We are inspired by G-d, but the words we communicate are our own. Psalms and Proverbs (David and Solomon) were written by those men in their own words, but inspired by their great love for G-d. Divine inspiration happens when a holy person meditates intensely -- resulting often in visions giving them a clear understanding beyond that of others. It is critical to realize that this is not speech with G-d (prophecy). Prophecy is a greater level of connectedness. Ergo when the bible says that the spirit of G-d rested on someone (in this case the messiah) it means they were divinely inspired.
Isaiah is saying that the messiah will have divine inspiration, and this will imbue him with wisdom, understanding and so on.
This connectedness to G-d does not appear as if by magic, descending from the sky -- it comes from within with a great deal of spiritual "soul searching." Matthew's claim is based on pagan ideas of gods, not on anything found in the T'nach, and certain not biblical (the Jewish bible).
What of the claim that this "spirit" anointed Jesus?
Matthew 3:16 - 17 does not mention anointing, and a good thing, too. To be a Jewish messiah requires a very special anointing which was never done to Jesus -- and to which he had no right (by virtue of his birth).
To be a Jewish messiah (priest or king) a person must either be personally anointed with a very specific type of oil (more on that in a moment) or be the uncontested descendant of one who was anointed with a very specific type of oil. There was a 600 year gap between Zedekiah, the last king of Judah, who was forced out by Nebuchadnezzar in the siege of Jerusalem (Y'rmiyahu / Jeremiah 32:4-5) and Jesus supposed birth. A 600 year gap = a contested lineage. Ergo Jesus would have to have been anointed with the special oil himself -- and he never was so anointed.
Quite simply the Hebrew word מָשִֽׁיחַ (moshiach) -- or "messiah" in English translates to "anointed one."
The noun מָשִֽׁיחַ (moshiach aka anointed one), in English "messiah" occurs 39 times in the T'nach. Thirty-four are nouns and the remaining five are adjectives.
Additionally there are other closely related words, such as מִשְׁחָה mish'ḥah (“anointment”) which occurs, for example, in the expression שֶֽׁמֶן מִשְׁחַת קֹֽדֶשׁ shĕmĕn mish'ḥat kōdĕsh (“anointment-of-holiness oil”)— this term is found twice in Sh'mot / Exodus 30:25 and again in Sh'mot / Exodus 30:31. שֶֽׁמֶן מִשְׁחַת קֹֽדֶשׁ shĕmĕn mish'ḥat kōdĕsh (“anointment-of-holiness oil”) is a special type of anointment with a particular oil for kings -- and this was never done with Jesus. Jesus was never a moshiach (messiah) -- and he did not have the "birth right" to be a rightful king of the Jews.
The Hebrew word is inseparable with the concept of the special oil in my last paragraph. This was a special mixture of spice and olive oil that was used for “anointing” of kings and priests. It is called שֶֽׁמֶן מִשְׁחַת קֹדֶשׁ shemen mish'ḥat kodesh (“Oil of Anointment of Sanctity”) in the passage of Sh'mot / Exodus 30:22-33 which gives the formula for making it and how to properly use it. . .
"G-d spoke to Moses, saying: 30:23 You must take the finest fragrances, 500 [shekels] of distilled myrrh, [two] half portions, each consisting of 250 [shekels] of fragrant cinnamon and 250 [shekels] of fragrant cane, 30:24 and 500 shekels of cassia, all measured by the sanctuary standard, along with a gallon of olive oil. 30:25 Make it into sacred anointing oil. It shall be a blended compound, as made by a skilled perfumer, [made especially for] the sacred anointing oil. 30:26 Then use it to anoint the Communion Tent, the Ark of Testimony, 30:27 the table and all its utensils, the menorah and its utensils, the incense altar, 30:28 the sacrificial altar and all its utensils, the washstand and its base. 30:29 You will thus sanctify them, making them holy of holies, so that anything touching them becomes sanctified. 30:30 You must also anoint Aaron and his sons, sanctifying them as priests to Me. 30:31 Speak to the Israelites and tell them, 'This shall be the sacred anointing oil to Me for all generations. 30:32 Do not pour it on the skin of any [unauthorized] person, and do not duplicate it with a similar formula. It is holy, and it must remain sacred to you. 30:33 If a person blends a similar formula, or places it on an unauthorized person, he shall be cut off [spiritually] from his people." Sh'mot / Exodus 30:22-33.
Thus Saul, David and Solomon were all messiahs -- anointed kings.
And it is just as clear that Jesus was not a messiah.
Jesus was not anointed with the שֶֽׁמֶן מִשְׁחַת קֹדֶשׁ shemen mish'ḥat kodesh (“Oil of Anointment of Sanctity”) in the passage of Sh'mot / Exodus 30:22-33.
Likewise Jesus was not an authorized person -- he did not have the tribal lineage to be a Jewish king, and he was not anointed by a prophet (as required by the bible).