Catholics certainly agree that all true Christians are priests (1 Pet. 2:9). We call this the "common priesthood."
Melchizedek is the only priest of the Old Testament who was not a Levite. He sets the stage for Christ as High Priest.
"Melchizedek brought out bread and wine; he was priest of God Most High. He blessed him and ...Abram gave him one-tenth of everything."(Gen 14:18-19)
Right after his blessing upon Abraham, Abraham's son Isaac was born and God's promise that he would father a great nation came alive (Gen 17:2). King David prophesizes Jesus as the new High Priest:
"The Lord said to my lord 'sit at my right hand' . You are a priest forever according to the order of Melchizedek" (Psm 110:1-4)
At the last supper Jesus served bread and wine (the first Mass) just as Melchizedek had done with Abraham (Gen 14:18). He said to the disciples "this is the New Covenant in my blood" (Lk 22:20), signifying, among other things, God's transfer of Priestly duties from the Levites to Jesus who was the "true priest with the others [disciples] being only his ministers" (Aquinas, Hebr. 8.4). That night Jesus washed their feet and taught them to be servants in their new ministry. He said "I have set you an example, that you should do as I have done to you" (Jn 13:15). Catholics believe that night, he conferred the ministry of the new priesthood upon them.
Within a short time, there was a structure to this new ministerial office with bishops, deacons, and priests (1 Tit 3:1). Peter was it's head (Mat 16:18, Jn 21:17).
In the Old Covenant, God ordained Aaron, the brother of Moses, to be a priest (Ex 38:21). He extended the spirit of Moses to seventy wise men and shared among the sons of Aaron the fullness of their father's Power. That's why Jesus called the priesthood "Moses Seat" (Mat 23:2). Their ministry passed from generation to generation through the bloodline of Levite family.