Alright I'm just going to drop my theory here on what the Fierce Deity is.
In order to do that, I'll present you a better understanding of the information in-game.
In the Japanese game, the kids wearing the bosses' masks ask Link to play hide and seek. It is a well known in Japan a game named kakure oni (隠れ⿁鬼?), or more commonly kakurenbo, that means chase the demon and it is the same as the hide-and-seek game that children in western countries play.
The kid wearing Majora's Mask is asking Link to play tag. In Japanese versions of the game tag, the player who is "it" is instead called the "oni." First you can see the Japanese text, then the direct translation, then the American version.
・・・だれも いなくなったな
...Everyone’s gone....
Everyone has gone away, haven't they?
オレと・・・あそぶ か?
Will you...play with me?
Will you play...with me?
オマエ もう、お⾯面 ないんだな・・・
You don’t have any more masks...
You don't have any masks left, do you?
じゃあ、別のあそびを しよう
Then, let’s play something else.
Well, let's do something else.
オニごっこ・・・か、いいな そうだ、それがいい
How about...tag. Yeah, that’ll work.
Let's play good guys against bad guys...Yes. Let's play that.
⿁鬼神の仮⾯面を もらった オニが かぶる オニのお⾯面? [C]でかぶってみよう
You got the Kishin Mask. Is it an oni mask that an oni wears? Try it on with [C].
You got the Fierce Deity's Mask! Could this mask's dark powers be as bad as Majora? Try it on with [C].
いいか・・・オマエが オニだ オニは 逃げるだけだ それでも いいよな
Good...You’re the oni. The oni just runs away. Even so, that’s fine. Are you ready?
You're the bad guy. And when you're bad, you just run. That's fine, right?
じゃあ
Well...
Well...
いこうか
Shall we go?
Shall we play?
The kanji for Kishin literally mean "Oni God". It translates to Fierce God or Demon God.
As you can see, just the American version depicts the thing to be bad and have dark powers, but there's nothing like that in the Japanese version.
The term "fierce god" or "fierce deity" (⿁鬼神) is a term referring to several deities in the Shingon Buddhist pantheon, such as Bishamonten or the Myō-ō, who are fierce and wrathful toward the negative forces that work against humanity's enlightenment, but are benevolent and supportive to humanity. They are seen as deities that work to defeat evil, a quality Link possesses in all his appearances, and which the Fierce Deity's Mask gives him greater ability to achieve. Kishin does not actually equate the subject with oni or evil at all; it merely states that the deity is as fierce as an oni. The term does not carry any connotations of evil in the Japanese language.
In Buddhism, wrathful deities are enlightened beings who take on wrathful forms in order to lead sentient beings to enlightenment. They are a notable feature of the iconography of Mahayana Buddhism and of Tibetan Buddhism, and other Vajrayana traditions in particular.
A wrathful deity is often an alternative manifestation of a bodhisattva or other normally peaceful figure, making the representations of all human vices and atrocities. True to their name, in Tibetan art, wrathful deities are presented as fearsome, demonic beings. While the deities have a hideous and ferocious appearance, they are not personifications of evil or demonic forces. The ferocious appearance of these deities is used to instill fear in evil spirits which threaten the Dharma.
Every other bit of information I could find on the Fierce Deity:
In the Beta version of Majora's Mask, also known as Zelda Gaiden, an Adult Mask of Link can be seen in the game. The sketches found in Hyrule Historia on Majora's Mask imply Link was supposed to transform into an adult at some point. However, this mask was considered useless because Young Link was able to hold all weapons. So instead, this mask is reforged into the Fierce Deity's mask.
The gossip stone says: "The
Fierce Deity Mask, a mask that contains the merits of all masks seems to be...
somewhere in this world..."
Ever since I saw that interview with this quote, it just added up even more;
"Whose soul is inside the Fierce Deity Mask?
Aonuma: The best I can give you is just a suggestion. The best way to think about it is that the memories of all the people of Termina are inside of the Fierce Deity Mask."
And both statements make sense. Because in order to represent all humans vices and atrocities, you need to complete the entire game and have helped every person in it. Doing so makes you able to get all the masks that consists their memories, as stated in game that masks are made from memories of that person. When you manage to get hands on all the masks, you will be granted the Kishin Mask; a mask that doesn't actually have a soul as it's the alternative manifestation of Link himself.
So, here's my thoughts on the subject.