As best I can elaborate:
The mastersword is originally the goddess sword created by the goddess Hylia to combat the arch-demon Demise. After the defeat of Demise, he binds himself into an eternal cycle of reincarnation. Demise is "dead", but his malice and contempt for Hylia and peace is eternally reincarnated. (It need not even be as Gannon!! It has taken the form of the Dark Sorcerer Vataii a few times!) Hylia, likewise, is doomed to eternal mortal reincarnation to combat this threat (in the form of Princess Zelda). This is because her power alone is not sufficient to defeat Demise, and in order to defeat him, she needed the triforce. Din, Farore, and Naru, the goddesses that created the world Hyrule is a part of, wisely chose not to let their token be used by divine powers; It can only be used by mortals. This is why Hylia, mortally wounded from her stalemate with Demise, concocted her plan to reincarnate as a mortal. (Zelda.) She best reflects the "Wisdom" portion of the triad, and it naturally gravitates to her when she incarnates. She incarnates whenever Demise incarnates, so in a way, Zelda is the herald of the coming of the disaster. The eternal champion is more a mythic figure that repeats as a focus between the battle between Hylia and Demise. In the first go around, his arrival is "manipulated" into being by the machinations of Hylia, as she needs a protector for her mortal incarnation so that her incarnation can perform her function to contain the calamity.
The master sword is a shard of divine power, and its manifestation changes as a result of any other divine influences it may encounter. In skyward sword for instance, it starts as the Goddess Sword, and goes through 2 physical transformations as it absorbs divine power from the 3 major goddesses. In other games, it receives additional powerups from other deities as well, further changing its physical form in appropriate ways after having done so. The sword is itself semi-sentient/alive, and can be considered an attending spirit in and of itself.
I would say that over 10,000+ years, by the time that BotW takes place, it has absorbed many sources of additional divine power to keep its embers hot-- (being cut off from a now less than divine creator, due to that creator being constantly reborn as a mortal, so that she can make use of the triforce of wisdom) and as such has probably seen more makeovers than Joan Rivers did.
This constant reincarnation was something I doubt the original goddess form of Hylia intended, which would explain why other divinities have had to pony up to intervene over the ages.