An Italian can correct me if I'm wrong, but I hypothesize that she's using the unusual form to emphasize that she's talking to a woman, specifically, herself.
Scatola is the theme song to the film/documentary Laura Pausini - Piacere di conoscerti, and she explained about the song, "Mi sono rivista adolescente, ho ripensato ai sogni che avevo e a quella che sono oggi, cercando un punto di incontro."
In the film, she imagines what life would have been like if she hadn't won San Remo and told her mom she felt like two people, the one that listened to her mom and avoided fame, and the one that listened to her dad and pursued music.
When she says, "We're the same thing (siamo la stessa cosa)," she's taking to her younger self before stardom, and means it quite literally.
That doesn't explain her finding "your" cell number and trying to reach you, BUT, Laura doesn't write her songs from scratch. She finds songs that resonate with her and adapts them to her personal circumstances (like La Solitudine: the song was written - she just had to change "Anna" to "Marco" and it became her story).
Like any good storyteller, maybe she left some of the original lyrics so the listener could interpret it a different way and see themselves in the song.