Lyric discussion by naughtmoses 

Cover art for Sister Blister lyrics by Alanis Morissette

While there may well have been an external object in Al's mind when she wrote this, she's working on a more internal level here primarily. Al understands internal conflicts and something called "borderline personality organization."

She has gone down this road before (see "Mary Jane" and "Break," for example).

Al is =very= aware of her internal conflicts and uses her writing to try to work through them so that she is less (unconsciously) "split in two" and more (consciously) "integrated."

Authors like Pia Mellody and Claudia Black, as well as Marsha Linehan and Glenn Schiraldi (the latter two at a more technical level), have written a lot about this topic.