The priesthood of Melchizedek was unique, because Melchizedek was unique - a one-off. It should not be surprising, therefore, that when God speaks of Christ as being priest forever in the order of Melchizedek (Hebrews 4:14 to 5:10), this is a unique, one-off matter.
Before drawing parallels between Melchizedek and Christ, here are a few quotes from Reformed Protestant sources as to its stance on priesthood, as an office in the Christian church, which will provide a foundation for grasping its conclusions on how many there might be in the priesthood order of Melchizedek.
"An alien concept is... the assigning to ministers a mediatorial work
as priests. Scripture asserts that Christians as a whole constitute a
priesthood (1 Pet.2:5). Scholars are agreed that a sacerdotal ministry
does not appear in Christian writers until the close of the second
century. 'Sacrifice is no part of the Christian ministry' (Hooker)."
(In Understanding Be Men, p1612, T.C. Hammond, Inter-Varsity 1997
Reformed Protestants are keen to point out the significance of Christ not being in the Aaronic priesthood, which came much later on than that of Melchizedek. This is expanded on in the notes in the NIV Study Bible, regarding Hebrews 6:20 to the end of chapter 8 where the ministries of Melchizedek and Christ are compared.
"Of particular significance is Melchizedek's holding both offices -
king and priest [Heb.7:1-10], one of the ways in which he prefigures
Christ... King of righteousness, king of peace, are Messianic titles
(see Isa.9:6-7; Jer.23:5-6; 33:15-16).
'Without father, or end of life - Gen.14:18-20, contrary to the
practice elsewhere in the early chapters of Genesis, does not mention
Melchizedek's parentage and children, or his birth an death. That he
was a real, historical figure is clear, but the author of Hebrews (in
accordance with Jewish interpretation) uses the silence of Scripture
about Melchizedek's genealogy to portray him as a prefiguration of
Christ. Melchizedek's priesthood anticipates Christ's eternal
existence and his unending priesthood." (NIV Study Bible, p1825)
The Study Notes then go on to show how Melchizedek was greater than Abraham, and that his priesthood was in effect before the Mosaic law was given. But with the Aaronic priesthood, the law was integral to that. Heb.7:11-17 expressly states:
"If perfection could have been attained through the Levitical
priesthood (for on the basis of it the law was given to the people),
why was there still need for another priest to come - one in the order
of Melchizedek, not in the order of Aaron? ...For it is clear that our
Lord descended from Judah, and in regard to that tribe Moses wrote
nothing about priests, And what we have said is even more clear if
another priest like Melchizedek appears, one who has become a priest
not on the basis of a regulation as to his ancestry but on the basis
of the power of an indestructible life."
The NIV Study Bible notes point out that the Aaronic priesthood was imperfect, but Melchizedek's was perfect:
"The announcement of the coming one who would be a priest for ever
(Ps.110:4) was written halfway through the history of the Levitical
priesthood, which could only mean that the existing system was to give
way to something better, for that one to come is priest for ever"...
Jesus is a perfect high priest for ever; so he is able to save
completely and for all time. His people will never be weithout a
priestly representative (see Jn.17; 1 Jn.2:1)." (Ibid. p1826)
This means that although the Bible does speak of Christians reigning with Christ as priests and kings (Rev. 7:9-15 & 20:4-6), it is neither in the order of Aaron nor of Melchizedek. That is because the Aaronic priesthood passed away with Christ's supreme, once-for-all-time perfect sacrifice of himself as the great high priest in the order of Melchizedek. Because Melchizedek was only one, and Christ is only one, their unique status means no other humans can approach that ministry. Christ alone carried it out. No more sacrifices are needed. This is confirmed in this final quote:
"The New Testament attributes Jesus' appointment as High Priest to a
higher order than the one prescribed under the old covenant: namely,
the Melchizedek priesthood, after the priest-king whom Abram
recognized as his superior in Genesis 14:18-20 long before the giving
of the law and therefore long before the Aaronic priesthood... This
makes Jesus the guarantor of a better covenant - Heb. 7:18-22. Thus,
priesthood and covenant are inextricably connected; a change in one
requires a change in the other. Or rather, since covenants cannot be
amended, an inferior one must become 'obsolete' in the inauguration of
the superior one (Heb.8:13). Christ's priesthood is therefore not
grounded in history, but in eternity; not at Sinai, but in the
heavenly Zion." (Pilgrim Theology, pp186-7, Michael Horton,
Summary of Answer: Christ alone is the one great High Priest in the order of Melchizedek. This priesthood is uniquely fulfilled in Christ, with no other literal priestly system on earth since his perfect sacrifice, let alone any re-enactment of sacrifices since then, or men claiming to continue Aaronic/Levitical positions. The role of Christians as stated in Revelation 1:6 is a spiritual function, all members of the Christian church being such ones designated as kings and priests unto God. This is spiritual, not literal. All members of the Christian church are "one in Christ Jesus", without distinction as shown in Gal.3:28 & Col.3:11.