Yep, meaning "the one responsible for my sister." which could be both, a masculine or feminine character.
I dunno, though, if in the novel made into film it was word-playing intended; perhaps it meant that younger Anna was (in the sense #1) "the person to be born to care of a valuable asset -since she was born to donate her bone marrow to Kate (that is dying for leukemia).
Or (in the sense #2), she is ''responsible" for her sister, although in that sense, if one watches the movie sees that both are responsible for the other, in a sense.
Still in sense #2, to contrast/compare with the saying "I am not my brother keeper" (from Biblical origins). That, at the time of the writting of the novel, would certainly draw people´s curiosity it (the novel).
from Cambridge. Dict.
I am not my brother's keeper. saying
used as a way of saying that you are not responsible for what someone else does or for what happens to them