The word "fair" has two possible meanings as you say. One has to do with skin, one has to do with life as a whole.
When we say someone has fair skin, we of course mean that their skin is pretty white. It is not a pejorative term like saying someone is pale. Instead, it is complimentary, meaning that their skin looks nice (obviously, this is only applicable to "white" people). The daughter is pretty critical of the way she looks and so she could be looking at her skin and being unhappy because her skin is not fair.
On the other hand, the word "fair" can refer to live in general and the daughter's place in life. She wants so badly to look some particular way but she can't. It's not fair, it's not right. In this sense, then, the word "fair" (or "not fair") is referring to her general dissatisfaction with her looks.