Jesus had already been baptized by John in the Jordan and been validated by His Father that He was the Son of God. This day in Nazareth does not suggest anything was added in Jesus' life and ministry. He had already been given the Spirit without measure, as John 3:34 states. He was simply communicating the fact that He was the one of whom Isaiah spoke. The time had come and He was anointed by God to fulfil His Messianic mission. Literally, Messiah and Christ both mean 'anointed.'
Jesus Christ was THE anointed One, however, Christians have Christ inside, spiritually. "Christ in you, the hope of glory." (Colossians 1:27) It is in this sense that Christians share the anointing of Jesus Christ. Some believers refer to the power of God or the Spirit of God residing in a Christian as the anointing. Scripturally, this is correct. "Now He who establishes us with you in Christ and has anointed us is God, who also has sealed us and given us the Spirit in our hearts as a guarantee." (2 Corinthians 1:21, 22) "But the anointing which you have received from Him abides in you." (1 John 2:27) This is not additional to the gift of Holy Spirit God gives at the time of the new birth.
Peter told the crowd on the Day of Pentecost to "Repent, and let every one of you be baptized in the name of Jesus Christ for the remission of sins; and you shall receive the gift of the Holy Spirit." (Acts 2:38) The Christian may choose to not utilize the gift God has given, but the gift resides within. It is "eternal life" as Romans 6:23 states, and "incorruptible seed" as 1 Peter 1:23 says.
Some believe that God pours on more of His Spirit for specific people or occasions. This may be due to a couple of sections in the book of Acts. "God worked unusual miracles by the hands of Paul, so that even handkerchiefs or aprons were brought from his body to the sick, and the diseases left them and the evil spirits went out of them." (Acts 19:11, 12) Paul and the believers in Ephesus were speaking the Word of God like wildfire. The intensity of faith was high among the believers. Jesus said in Mark 16, "These signs will follow those who believe."
This was not an extra outpouring on Paul, but a time of great community believing. This is similar to the time recorded in Acts 5, when all of the apostles were moving with the power of the Holy Spirit. "And through the hands of the apostles many signs and wonders were done among the people ... so that they brought the sick out into the streets and laid them on beds and couches, that at least the shadow of Peter passing by might fall on some of them." (5:12, 15) These are not accounts of special extra powers. The reason for these upsurges of miracles and healings was the rise in the level of faith. Faith pleases God and gets results. (Hebrews 11:6) If God gave more to one person than another, He would be a respecter of persons, and Romans 2:11 specifically states that he is not.
Christians don't need anything extra. Colossians 2:10 declares that Christians are complete in Christ. Complete means lacking nothing. Philippians 2:13 teaches that God "Works in you both to will and to do for His good pleasure." The word translated 'works' to mean that God energizes within. This same word is used in 1 Corinthians 12:6: "And there are distinctive varieties of operation [of working to accomplish things], but it is the same God Who inspires and energizes them all in all." (Amplified Bible)
The anointing of the Holy Spirit is another term for God's gift, like the terms Comforter(KJV), Helper (NKJV) or The Spirit of Truth that Jesus spoke of in John. Jesus also called it The Promise of the Father in Luke 24:49 and Acts 1:4. Every Christian has the anointing of Christ within.