It means that her more is like another piece of her. Here's a different example: Learn new languages is a part of WHO I am. <- it has a similar meaning=> if I don't learn new languages I would feel NOT myself. So, she is saying her mom is a part of her life and without her mom she won't feel the same.

Such a part of me - just like you. There's a connection between you and her. hmm, this is only my opinion. Hope that it helps. :)

It is figurative and means something that is very familiar or necessary.