The term ‘Order of Melkizedek’ have a special meaning. First one has to understand what it means to be a priest, who is Melkizedek, then what it means to be of the Order of Melkizedek.
This is how I understand it. Melkizedek is presented as having no father nor mother. Yet he is a man. How is it so? When the account was written, hebrews were comparing the priesthood of Melkizedek with the priesthood of the Levites.
Levites where people who were priest simply because they were of the tribe of Levi. In other words. One could not be a priest unless he is from the tribe of Levi. So it was important who your mother or father was because you were to be traced back to your patriarchs.
But the priesthood of Melkizedek was different. It was the primordial priesthood. Or what you may call the natural priesthood before people are fallen into sin.
From Eden, Man is called to be a priest. That is someone who offer sacrifice for his brethren and family. The tree of knowledge was meant to be a sacrifice; the only tree which was not to be eaten.
After the fall of Adam, humanity went worse until it was destroyed by the flood and Noah was spared. Noah was the new family chief of mankind. This means he was also a priest-king for mankind. As a priest he was to offfer prayer on behalf of all his family (wife and chidreen meaning the whole humanity).
But at his death who was to replace him in this activity??? His first born son. God has organized mankind in unity from the beginning. We are all united from the fact that we have an original first parent. And we were not going to be divided after his death. No. The first son always was called to replace his father unless he was found not worthy to do so in which case his little brother was considered. This is how God meant it which is why we grow up learning that we have to obey our parents and our elder brothers.
So here, the priesthood does not depend on your parent, but on the order of your birth. This is whether you are first or not in the family. Every first born boy is meant to be a priest for his father’s family. To offer sacrifice and prayers on behalf of his little brethren.
So this lead to who is Melkizedek. Melkizedek is actually a title not a name. It means ‘king of righteousness’. So this person was a first born of the whole humanity after the flood. He was meant to rule and teach to his brethren the ways of righteousness. His name was not mentioned, may be because it was obvious who it was at that time?! Most Jewish and Catholic theologians believe that it was Shem, the first born of Noah.
Naturally, if humanity had remained in God’s grace after the flood, Noah was to rule it because he was the father. So as the father of the new human family he is the ruler (King, and every father is king over his house), and the priest (offer prayer and sacrifice for his family - job for every father). And after death his first born son was to replace him, and so on, and no body would have complained because this is the family way, this is the way of love, the way of God.
It is in this sens that Jewish are chosen people of God. Because they are from the first born of humanity. They are in the order of the first born. So as first born, they are naturally responsible for all humanity. And God stayed close to them and their fathers in a particular way because of this fact. God is always close to all humanity, but the expected semites (shemites, particualarly hebrews…jews) to set an excellent example of the rest of humanity because humanity naturally look up to them. Early human society new them and recognized them as first borns.
Jesus is of the order of Melkizek because not only he is the first born of Mary, a jew, but because he is the first born of God himself.
Catholics are in the order of Melkizedek because their priests are partakers in the priesthood of Jesus. But also, I see this order in the hierarchy of the Catholic Church.
Peter was the first born of the Church by his confession of faith. Therefore, he was put in charge of his brothers to confirm them in the faith. Catholic Church is the new people of God, a new society, governed by the Pope who is no other but a successor of Peter. All Popes are bishops in the line of Peter.
The Pope and the bishops with him and all catholics are basically what humanity would have looked like if humanity had not sinned after the flood. As Christians we are all priests. The Church is a society of people who are a royal priesthood. This is just what I was saying. Everyone is supposed to be a priest-king from the nuclear family to the highest hierarchial member of humanity. So fathers are priest-kings in their families. But for many families we have a bishop, and then a patriarch, and then a Pope.
God bless