While I do like the idea of Wilds animal form/spirit being a wolf or fox more so fox because it fits with the theme of Time Twilight Wild while detailing how Wild isn’t a descendent/ direct successor of Twilight but having the familial relationship and the mystery and slighness of a fox is more in tune with wilds fighting style but that whole other can of worms a phoenix is just much for fitting for Wild as a character.
First is the obvious: Wild dying and coming back to life, burning one’s self to fly higher, rebirth after burning/ dying very very much is representative of Wild as a character. In eastern mythology/ religions specifically Chinese, the phoenix is related towards balance and moral integrity. In which loses all your memories would and made Wild have to really think about what his morals would be with out the direct influence of the society he grew up in. And when pared with a dragon it symbolizes equilibrium.
Note there is a bigger discussion to be had about Wild and Flora’s relationship from pre Botw to post totk and how they’ve been thematic foils of each other since they’ve been children.
Further more the modern context for the phoenix is one survival and personal growth something that has been integral to all four games ( yes even AoI bc the existence of Clink as a “stand in” for link does give us more insight on his character)
Wild as a character is such a complex character especially after totk because of everything that has happened that specific characteristics/ details get washed out or hidden, but the concept of rising from the ashes and his stubbornness is something is persistent though out his whole life.
