Who is Melchizedek?
There is significant evidence that Shem is Melchizedek. Although Japheth was the oldest of the three sons of Noah, whenever they are mentioned in scripture, Shem's name appears first. Let’s look at it, realizing, of course that some of it is speculative.
Shem lived to be 602 years old
Abraham was 140 years old when Shem died.
Shem ruled over his progeny and their descendants along with Ham and at least some of his descendants, thus he was a king. (Gen. 9:26) He is said to have been the king of Salem, which could have been a short name for Jerusalem.
It is supposed that patriarchal law included a priesthood, which was appointed by God. Sacrifices were made in Adam’s day for we know God was not pleased with Cain’s offering, thus suggesting that God required an animal sacrifice and Adam would have been the only possible mediator or priest. If Abraham paid tithes to Melchizedek, such would have been a matter of continued support of his priesthood as it was for the Aaronic priesthood. Jacob's vow, (Gen. 28:22) along with the account of Genesis 14:18-20 support the idea of tithes commonly paid to priests of that era.
So, if the man called Melchizedek is Shem, the name is then used as a title, not an actual name. (much as “pharaoh” was used for Egyptian rulers) This title would seem to fit the biblical account of the Godliness and righteousness ascribed to Shem. That name is translated “king of righteousness” The name in similar to another king of Jerusalem, Adoni-tsedek, translated lord of righteousness. (Josh. 10:1)
Hebrews 7:3 said he was made “like” the son of God, not that he “was” the son of God. Some do suppose he was Jesus himself, but that does not make much sense from this statement. It also follows that if Melchizedek was Jesus, he could not have been king of Salem, because Jesus said… “my kingdom is not of this world”.
Souidas (author of a tenth century historical dictionary/lexicon) said Melchizedek ruled in Salem 113 years. Josephus and Philo both speak of him as a man well known, the same way Moses described him. Jerusalem was about 20 miles south of Hebron, where Abraham settled his family, so word of the rescue of his family could easily have reached the king. Some objection is made, saying Jerusalem was a Jebusite city, (Canaanite), but it was formerly ruled by Shemites, so it follows that as Israel was in and out of idolatry, there could well have been a period under Shem, in which Jerusalem worshipped the one true God. During that period Abraham conquered the 4 kings and rescued his people and paid tithes to Melchizedek, king of Jerusalem and priest of God.
The difficulty with this understanding, is He was “Without father, without mother, without descent, having neither beginning of days, nor end of life; but made like unto the Son of God; abideth a priest continually.” (Heb. 7:3) That this man was a historical figure, recognized as having the dual offices as priest and king in the Genesis record, seems well understood from Jewish tradition. The only plausible explanation is that among the Jewish archives, in which genealogies were carefully preserved, no record of his family or priesthood was found. So, the Hebrew writer points out this righteous man, surrounded by Canaanite nations, was he who held the dual offices of priest and king as a type of Christ from the age of Abraham. The record showed he is revered as superior to Abraham, thus superior to Moses and Aaron as well. His priesthood abides continually through Christ, being from the order (patriarchal order) of Melchizedek.
If Shem is Melchizedek, it is possible that taking the title as a name would have removed from genealogical records his ancestry from Noah, especially since it is also possible no ancestral records survived the flood and the separation of his brothers from him over a century, left Abraham’s contemporaries knowing him only as is recorded in Genesis. (as described by Moses many centuries later). OR...God could have changed his name, as he did Abram and Saul, with the express purpose of obscuring his ancestry to establish a type to be used for the very purpose the Hebrew writer used it.
Albert Barnes concluded…”so far as the record goes, it is just as it would have been if his priesthood had neither beginning nor end. It was inevitable, therefore, that those who read the Psalm, (Ps. 110:4) and compared it with the account in Ge 14:1-24, should come to the conclusion that the Messiah was to resemble Melchizedek in some such points as these--for these are the points in which he differed from the Levitical priesthood;”
Aaron's ancestry from Levi was well known, as were his descendants, succeeding him as priests. Since Jesus was from the human lineage of Judah, he could not have been a priest under the Mosaic Law. He was a priest under the Patriarchal Law, that the Jews recognized as their seed through Abraham and would have to acknowledge the superior claim to the priesthood of Melchizedek.