How does Annie John relate vanity to her quest of identity ?

For most of her life, Annie's identity has been intimately bound up with the extraordinarily relationship she has with her mother. When she hits puberty and their relationship breaks down, she has to forge a new identity completely from scratch. Throughout the story, Annie is a pretty willful child with a strong sense of her own worth. None of the relationships she develops after the breach with her mom endure in any way. This is not simply because she is looking for something she had with her mom that is now irrevocably lost; it is also because she is fundamentally a vain person, believing herself to be much better than everyone else.
Annie's deep, loving relationship with her mother kept her vanity in check. Although they were more bosom buddies than mother and daughter, there was no doubt who the senior partner in the relationship really was. Annie's mom is very old-school when it comes to parenting. Her word is law, and the house is her domain. As long as Annie does as she is told, all will be well. God help her, though, if she ever steps out of line. Thank goodness, then, that they get on famously.
After Annie hits puberty, all that changes. Mom distances herself because she wants Annie to grow up and start acting like a lady. Annie is shocked and confused. Apart from anything else, she kind of thought that she was almost an equal in the relationship, her mom's "Mini Me," as it were. After all, they used to hang out together, go shopping, and even wear the exact same outfits. (They even have the same name). Annie already felt like a grown-up. Now she realizes that her mom thought of her as a little kid all along, and it hurts like hell.
At the same time, the breakdown of the relationship with her mom gives Annie the opportunity to let her real personality show. Now she can cut the apron strings, emerge from her mom's shadow, and start living like a real individual. However, there is really nothing to show; Annie's life has been so closely linked to her mom's identity that she has no true identity. The separation from her mom has left a gaping void in her soul which Annie somehow has to fill.
She really does not know how to do this. She throws herself into all kinds of adventures, desperately trying to cobble together something approaching an identity. First, Annie befriends Gwen. It is clear that she is looking for a mother substitute, and, initially, it seems that she has found one. This does not last long. The problem is that this is no relationship of equals. Neither was the relationship between the two Annies, but Annie John genuinely thought it was. There is no ambiguity this time, though. Annie is the mom and Gwen is the giggling, immature, little girl in the relationship.
However, we should not be too hard on Gwen. Annie's growing vanity is primarily responsible for their relationship not working out. She think she is a grown-up and wants to be with someone who makes her feel like a grown-up, just like her mom used to do. Gwen cannot do this. On the contrary, she acts as a reminder that Annie is still only twelve years old and just a kid.
This definitely is not a problem with the Red Girl. She has maturity way beyond her years. She shows that if you want to be considered as an adult, you have to act like one. You can gamble, you can steal, you can do whatever you like. After all, that is what adults seem to do all the time. There is no attempt by Annie here to replace her mom's affection. That ship has long since sailed. Now, she sees relationships purely in terms of self-gratification, as an opportunity to display her growing vanity and assuage her elevated sense of who she thinks she is. To hell with the consequences.
No one can ever truly match up to Annie's super fussy standards. She is the most important person in her own world. For a while, she thought her mom was amazing, but that is ancient history. Additionally, Antigua is such a little island, far too small a place to contain Annie's ever-growing ego. She needs a bigger stage to indulge her vainglorious visions. As a result, she leaves for England to work as a nurse. Not only will this allow her to indulge her morbid obsession with dead bodies, it will also give her the chance to be someone, a chance to create an identity in keeping with the superior self-image she has been attempting to develop ever since she fell out with her mother.