It’s much older than Vatican II.
First, here are some excerpts from my first book on the Mass that explains what priests, prophets, and kings have to do with baptism and Jesus:
After God reveals this name to His people, we find that many of the prophets and priests and kings of Israel receive a name that refers to this revealed covenant name of God. The prophet Isaiah relayed so many striking prophecies of Christ and His Passion; his name means “salvation [is] of
Yah[weh]” (“Isa
iah”). Jeremiah prophesied the restoration of Israel by means of a new covenant with God; his name means “
Yah[weh] will raise” (“Jerem
iah”). During a time when Israel was divided in two and the people worshiped false gods, Elijah came to King Ahab and revealed by many miracles that
Yahweh is the true God, not just of Israel but of all nations and all men; his name means “
Yah[weh] is God” (Eli
jah).
God commanded Israel to consecrate priests, prophets, and kings to Him by anointing them on the forehead with oil.
We enter into this three-fold office of Christ – as priest, prophet, and king – in our Baptism: after we are washed in the water, we are anointed on the head with oil.
The offices of priest, prophet, and king are fulfilled in Jesus Christ, and we are baptized into Him. We all share in some way in these offices, although the manner in which the ordained ministers of the Church (especially bishops) do so is very different from the manner in which the lay and religious members of the Church do.
Priest - offering sacrifice. There is the baptismal priesthood, by which we all share in Christ’s priesthood, enabling us to offer ourselves as living sacrifices to God, and to join ourselves to the offering of Christ in the Eucharist, offering the Eucharist with the ordained priest. There is also the ministerial priesthood, a deeper participation in Christ’s priesthood.
Prophet - teaching. All of us are called to spread the Gospel, whether by our lives or our words. The ordained are specially called to teach, and thus they can read the Gospel and preach the homily at Mass; bishops are the official teachers of the faith in their diocese.
King - serving. Christ is the perfect King who truly loves His subjects: He died for them! The kingship of Christ is one of service. When we perform works of mercy and charity, we are serving others as Christ does. Sometimes this service does require what we would consider to be “ruling” – setting down laws and disciplines for our well-being – and that belongs to the ordained ministers to carry out, particularly pastors of parishes and dioceses.