In modern Catholicism, archangels Michael, Gabriel, and Raphael are all considered saints in the formal sense. For everyone saying "but we don't call Gabriel a saint," have a Google at "St. Gabriel parish" and "St. Raphael parish." Catholicism is built on two poles: Scripture and tradition. While Michael and Gabriel are mentioned in Scripture (and so is Raphael, in this case--Tobit is in the Catholic Old Testament even if Jews and Protestant Christians reject that book), it's tradition that affords them sainthood.

The process of canonization--formal, official designation of individuals as saints by the Church magisterium--is a high medieval invention. But both before and after that, recognition of saints was a local phenomenon that might even begin during the prospective saint's lifetime. Christians recognized certain people as exceptionally holy, for a variety of reasons not all of which we would equate with "holiness" today, and thus closer to God. In particular, close to God in ways that they could request divine aid on behalf of other people--namely, those who venerated them by praying for their help, those who visited their shrines, those who donated money to churches and monasteries dedicated to them. Saints were believed to work miracles in life and act as intercessors with God after death, speeding good Christians through purgatory into heaven.

Once canonization was in play, there was a certain weirdness where supporters of a candidate for sainthood would try to assemble the "application package" for a canonization process on one hand, but on the other, actively aid the cult (cultus, or organized devotion) of the person as a saint in their own region. So a lot of medieval people we call "St. ___" are technically not. A great example is Barbara Newman's 1987 book Sister of Wisdom: St. Hildegard's Theology of the Feminine. 12th century abbess and visionary prophet (and everything else) Hildegard of Bingen was not actually canonized yet when Newman published her book, but for all intents and purposes, Catholic laity treated Hildegard like a saint.

As canonization settled into place as a Church institution, they basically grandfathered in all pre-existing recognition of saints. Mary Magdalene was never canonized; she's a saint. Martin of Tours, never canonized; he's a saint. And so on. Christians prayed to them for intercession with God in life and in death. The blanket acceptance of early saints eventually posed some problems for the (by then) Catholic Church, as they realized some of the most popular saints like Katherine of Alexandria and Margaret of Antioch could never have actually existed. (Their current status is...complicated.)

Michael, Gabriel, and Raphael all had cults of veneration that predated the introduction of canonization. However, there's no question that Michael was the most popular by far. An important reason for this was late antique and medieval tension over the potential power of angels as agents of magic. The Church worried about people invoking angels, just like demons, to complete ungodly sorcery. Michael, however, was considered unimpeachable. There is lots of evidence for official Church support of his cult from quite early--the Leonine Sacramentary from the seventh century (the link to Pope Leo is modern nomenclature) contains a liturgy in his honor. Very important early medieval theologian Rhabanus Maurus wrote hymns in praise of Sanctus Michael. His shrines were especially associated with mountains/high places--you know Mont-St-Michel in France, probably. Richard Kieckhefer suggests he was something of a "sky god" in comparison to Mary as an "earth goddess," with their cults skyrocketing in popular in tandem over the course of the later Middle Ages. Michael, like Mary, was a boundary-guardian, there between life and death, there at a Christian's deathbed.

Eamon Duffy reproduces one "magical prayer" from a popular prayer book in late medieval England that shows the prominence of Michael, but pulling the other two archangels with him (technically it):

God be favorable to me a sinner, and be my guard all the days of my life. God of Abraham, God of Isaac, God of Jacob, have mercy on me, and send to my aid Michael your Archangel, that he may keep, protect, and defend me form all my enemies, visible and invisible.

Holy Michael, the archangel of God, defend me in battle, that I may not perish in the dreadful judgment. Archangel of Christ, by the grace with you have merited I beseech you, through our only-begotten Lord Jesus Christ, draw me today and always from deadly peril. Holy Michael, Holy Gabriel, Holy Raphael, all holy angels and archangels of God, hasten to help me. [...]

Behold + the cross of the Lord, begone you enemies. The lion of the tribe of Judah has conquered; root of David, stem of Jesse, saviour of the world, who have redeemed me through your cross and blood, save me, help me, my GOd. Agios, Agios, Agios. Cross of Christ protect me. Cross of Christ, save me. Cross of Christ, defend me from every evil.

I don't have the Latin, but I suspect "holy" is the Latin sanctus, which can mean holy or saint.

Essentially, Michael, Gabriel, and Raphael became saints because Christians treated them like saints. Michael, considered the least ambiguous or rather ambivalent in terms of potentially being invoked for black magic, had the most central support from the Church--but also took on the tasks most important to medieval Christians, a guardian in their quest for salvation.

The English word “saint” derives from the Latin “sanctus”, meaning “holy”. So “saints” are just “holy persons”.

Being an Eastern Orthodox Christian I’m more familiar with Greek; in Greek it’s the same: a saint like St James would be “o άγιος Ἰάκωβος”, literally “Holy James” or “the Holy One James”. Similarly archangels like St Michael are referred to as “the Holy Archangel Michael” (“o άγιος αρχάγγελος Μιχαήλ”). There’s no difference between the words for “holy” and “saint” like in English.

So St Michael isn’t a “saint” in the sense of a human being who is recognized for achieving holiness, but as one of the chief angels of God he is certainly holy, and so referred to as St Michael.

You might want to try /r/academicbiblical