A high priest would stop becoming high priest upon his death.
Since, as the scripture mentions, Melchizedek had no record of his death he remains high priest as no note of his death was ever recorded. Jesus is high priest only in the sense that he remains high priest forever.
They cannot be the same person since the scriptures makes clear that Jesus was another high priest, if Jesus was the same high priest as Melchizedek then the Bible writer would NOT have called him another high priest like Melchizedek but rather the same high priest.
(Hebrews 7:1-3) "..For this Mel·chizʹe·dek, king of Saʹlem, priest of the Most High God, met Abraham returning from the slaughter of the kings and blessed him, 2 and Abraham gave him a tenth of everything. First, his name is translated “King of Righteousness,” and then also king of Saʹlem, that is, “King of Peace.” 3 In being fatherless, motherless, without genealogy, having neither a beginning of days nor an end of life, but being made like the Son of God, he remains a priest for all time.. If, then, perfection was attainable through the Levitical priesthood (for it was a feature of the Law that was given to the people), what further need would there be for another priest to arise who is said to be in the manner of Mel·chizʹe·dek and not in the manner of Aaron?..".
Christ was another high priest like Melchizedek, thus, he was not Melchizedek.