"You loved righteousness and you hated wickedness; therefore G-d, your G-d, anointed you with oil of joy from among your peers." T'hillim / Psalm 45:8 (7 in Christian versions)
First let's discuss the special oil which debunks this particular claim that Jesus was specially anointed, and then discuss the "holy spirit."
Quite simply the Hebrew word מָשִֽׁיחַ (moshiach) -- or "messiah" or "christ" (taken from the Greek) in English translates to "anointed one." The noun מָשִֽׁיחַ (moshiach) appears 39 times in the T'nach. Thirty-four are nouns and the remaining five are adjectives. This is mentioned because most Christian translations only use the word "messiah" in Daniel 9 or sometimes T'hillim / Psalm 2 -- the other 30+ instances of the word are NOT translated as "messiah" -- misleading readers into thinking there is only one messiah when in reality every properly anointed Jewish priest and king was (is) a messiah.
One could even properly call Queen Elizabeth II, the current queen of England, a messiah for the British. She was properly anointed according to the laws of her own country.
The Hebrew word, when used to speak of Jewish kings and priests, requires a very specific oil be used for the anointing of a priest or king (if there is a dispute on the lineage). The Jewish messiah is thus inseparable with the anointing of these people with a very special mixture of spice and olive oil that was used for “anointing” of kings and priests.
This anointing oil is called שֶֽׁמֶן מִשְׁחַת קֹדֶשׁ shemen mish'ḥat kodesh (“Oil of Anointment of Sanctity”) and the formula for it is found in the passage of Sh'mot / Exodus 30:22-33 -- pay special attention to this passage, because Jesus was never anointed with this special oil -- the ONLY oil that can properly annoint a Jewish king or priest:
"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 Jesus was not.
Matthew 3:16 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 -- G-d clearly states in Sh'mot / Exodus 30 that to be an anointed one (messiah) a very specific oil must be used and Matthew 3:16 and Hebrews 1:9 never claims that this oil was used on Jesus.
Hebrews 1:9 even QUOTES the line from T'hillim / Psalm 45:8 -- but "so what"? The facts are the facts: David was an anointed king and Jesus was not. Anyone can make a claim -- the proof is in the fulfillment not in the claim.
There are Biblical regulations for anointing Jewish kings, which Jesus did not satisfy. The oil used to “anoint” a messiah as quoted above (Sh'mot / Exodus 30:22-33) -- which was not used for Jesus is one requirement he failed. Jesus was not anointed by a recognized prophet (Shmuel Alef / 1 Samuel 10:1), which is another requirement. Also Jesus was not anointed on the correct part of the body -- "You shall take the anointing oil and pour [it] on his head and anoint him." (Sh'mot / Exodus 29:7).
What about the "holy spirit" anointing Jesus?
A person cannot be “anointed” (in the sense that this word is used in the Scriptures) with water, or with a dove, or with “holy spirit”, or with anything else apart from the compound of spices and olive oil that is specified in the passage Sh'mot / Exodus 30:22-33.
On to the second point -- there is no "holy spirit" or "holy ghost" in the sense of an entity -- a part of a trinity god. There is only one G-d (D’varim / Deuteronomy 6:4). The term "holy spirit" (in which "holy" is an adjective) never appears in the T'nach even once. The term does appear in Rabbinical writings -- but it is nothing like the Christian concept.
Judaism, relying on the T'nach, holds that G-d is One and Unique. The notion of a dual or triune godhead is totally rejected.
G-d is not a spirit.
G-d is not a physical entity (He has not physical form, He is not a man).
It would seem that this Christian “holy ghost” (“holy spirit” of the trinity) is based on a misuse of the Hebrew term Ruach HaKodesh (mistranslated as holy spirit) But Ruach HaKodesh doesn't mean "the holy spirit.” “Holy Spirit” would be הָרוּחַ הַקָּדוֹשׁ ha-ru'ah ha-kadosh (ha meaning “the”) or perhaps הָרוּחַ הַקְּדוֹשָׁה ha-ru'ah ha-k'doshah (because רוּחַ ruach can be both masculine & feminine – all nouns in Hebrew must be either male or female).
קוֹדֶשׁ (kadosh) is a noun ("holiness" or "sanctity") and literally means to be separate. The term רוּחַ literally means “wind” and is the lowest level of the soul – the life force.
רוּחַ הַקֹּדֶש / Ruach HaKodesh should be translated as "a spirit of [the] holiness" or "a spirit of [the] sanctity". Meaning it is a state of holiness in a human being, NOT a mythical G-d of spirit.
Let me repeat that – it is a state of holiness in a human being, NOT a mythical spirit of G-d.
So what does it mean? What is a state of holiness in a human being?
One inspired by G-d.
A state of being of a person.
Inspiration being a level below communication with G-d (prophecy).
Think of it as a kind of sub-prophecy or divinely-provided intuition.
A state less than that of prophecy (direct communication from G-d).
In summary: Jesus was never properly anointed as a messiah (king), a process which requires a very specific type of oil be applied by a known prophet. There is no "holy spirit" entity which could have anointed Jesus as a messiah -- because "G-d spoke to Moses. . .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. 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 30:31-33.