Family rules everything in Persuasion. It determines where you live, what you eat, whom you see, and what you talk about. Separating oneself from one's family is a necessary step to forming an independent identity, but few characters manage to take this step, since family connections are such a huge part of their position in the world. The fact that the protagonist's family doesn't care for her much makes her life at home difficult, but also helps her to develop her own ideas as to who she is and what she wants.
Anne's relationship with her family determines her identity: they control where she lives and who she meets.
Despite Anne's family controlling her physical existence, her independent mind means that she has an identity separate from them.