Screenrant has an elaborate animal symbolism page about the movie and their take on the Crane is:
Fei Fei encounters a crane again at the end of the movie, which takes place during the next year's Mid-Autumn Festival. Cranes also symbolize longevity and immortality, so there's a strong possibility that this particular crane could represent the spirit of Fei Fei's mother living on. The crane appears to look directly at Fei Fei and even smiles at her before flying off into the night. It then joins an entire flock of other cranes, representing the idea that the spirits of those loved ones still continue on, even after they have passed away.
Also from Looper
In an interview with Slashfilm, Glen Keane said, "It was ... a very important moment, when I first storyboarded that, Fei Fei was talking to her mom. And then we took all of that dialogue out and it played so much better. And her mother makes our appearance there, in that crane in a sense." Though Keane doesn't outwardly state that Fei Fei's mother is the crane, the crane is present at both the start and conclusion of Fei Fei's emotional journey. Based on the director's testimony, paired with these themes, it is heavily implied that the crane is Fei Fei's mother — or at the very least, closely connected to her spirit.
So Crane is implied to be (the spirit of) her mother.