Salero literally means "salt shaker". It's an unexpected and humorous word choice in this context, making it memorable and intriguing.
In the song, the line "no soy salero de nadie pa' que seamos tres" (I'm not anyone's salt shaker for us to be three) is a clever and colloquial way of saying, "I'm not a third wheel" or "I'm not here to be part of a love triangle." It implies that the singer refuses to be an accessory or an extra person in a relationship that already involves her partner and their past lover.
Yuridia spends this one talking to her boyfriend and his ex as if they were the couple and she were the outsider. She had made her peace with history: "Te acepté con tu pasado, pues todos tenemos uno" (I accepted you with your past, because we all have one). Then the history started answering back. "Pero tu pasado y tú, por las noches, se escriben" (But your past and you, at night, write each other), and the woman he swore meant nothing now gets "entrada libre pa' tu piel" (free entry to your skin). The line that does the most damage is the smallest one: "Y no soy salero de nadie pa' que seamos tres" (And I'm not anyone's salt shaker so that we can be three). A salt shaker is the thing that gets passed around the table, and she is not it.
What makes it land is how calm she stays. She wishes them well in the same breath she walks out, "Ojalá que duren y que seas feliz" (I hope they last and that you are happy), then adds a closing message for the two of them that is about as blunt as Mexican Spanish gets. She has stopped treating him as one person. He and his past are a couple now, and she is the one leaving.