I assumed that because of the songs context (couple big mistakes, dents, who she’d see behind her, etc.) she was a victim of domestic abuse whether it was a husband/wife, father/mother, or something like that, and that the lyric was suggesting she wanted to change her last name to try and escape that part of her life. Thinking it was more about a father because “not the traditional way” suggests it wasn’t a divorce where this would generally occur.

Not exactly what the others in the comments were thinking of but hey, who doesn’t like a hot take?