Now, when they encounter Melchizadek, King of Salem, he is explicitly depicted as being mysterious; and it’s not surprising that he is frequently associated with the long-lived patriarchs, to the point that many Jewish sources from ancient times have taught that he was actually Noah’s son, Shem, still alive after all these years. Many early Christian interpreters taught the same thing.
This is really a side issue, but I wouldn’t say there were ‘many’. (I could only remember Sts. Ephrem, Jerome, and Nicholas of Lyra explicitly making this identification.) The thing is (you might probably already know this, but to explain for people who don’t know), the ages of the patriarchs is one of the trickier parts of Genesis,
since the surviving versions just don’t agree exactly on how old they are supposed to be. The so-called proto-Masoretic Hebrew text (the ancestor of the Masoretic Text), the Greek Septuagint (based on another Hebrew version different from the proto-Masoretic text) gives yet another, and the Samaritan text (used by Samaritans today, yet another version different from the MT and the Septuagint) all give different ages.
Jews who used the proto-MT could claim that Shem is Melchizedek, since in the version they use, Shem would have been still alive at the time of Abra(ha)m. But that’s not the case with many early Christians, many of whom used the Septuagint or a derivative version, since in their version of Genesis, Shem should have been dead in Abram’s time. Now, those Christians who are familiar with the text Jews used (such as Jerome or Nicholas of Lyra - both knew Hebrew) or were using a non-Septuagintal version of Genesis (Ephrem) did know of and made this claim, but there’s another dilemma: how will they square this belief with what is written in the Septuagint? One writer got around it by suggesting that while Melchizedek was not Shem himself, he
did bless Abram in Shem’s place: so in a way, he can be called ‘Shem’ because he acted as the latter’s proxy.
The thing is, this identification of Melchizedek with Shem wasn’t prominent historically among Christians (the Haydock Commentary even explicitly
rejects this identification), until recently, when people like Scott Hahn (on the Catholic side) began to espouse this idea. Some of these folks will even say that this is ‘Jewish tradition’, but this is a half-truth: this is just the latest variety of the various ancient Jewish ideas about who Melchizedek actually was. As you said, he was a very mysterious figure. (In fact, it’s even likely that the ‘Melchizedek is Shem’ belief arose as a reaction
against the Christian tendency to use Melchizedek, who is often presented as a non-Jew, as a prooftext for the idea that circumcision was not as important as Jews made it out to be - after all, if Melchizedek was a priest of God, how come he did not undergo circumcision? In other words, by making Melchizedek to be one of the distant ancestors of the Hebrews, the Jews ‘domesticated’ him and made him a less problematic figure.)