It's the song that Dagger/ Garnet sings throughout out the game, that hum song and the song that she sings in Dali ( or rather have some random woman do la-la-la's in the tune of the song ) And the song that Zidane mentions is beautiful in Lindblum and the song he asks to hear again.

Garnet says that she doesn't really know where it comes from or where she heard it last; but she says that it calms her down and keeps her cheerful when she's alone.

Zidane clearly remembers Garnet's song and sings it when he's traveling back to Alexandria castle to survive and again, remind him that he's not alone.

It clearly started out as a Summoner Village song, Dagger managed to remember it- making it her song and then it becoming Zidane and Dagger's song. Nothing to do with FF IX being the last game of the PlayStation era, or anything like that. Just a song symbolizing that " we parted, so when will we see each other again " ? Part of the FF IX ending and Zid's and Dagger's love relationship.