Susan Stringham is an aging woman who wants to bask in the reflected glory of the beautiful young Milly Theale. As the narrator tells us, "[she] had lost first her husband, and then her mother, with whom, on her husband's death, she had lived again; so that now, childless, she was but more sharply single than before." (22.214.171.124). Susan is tired and lonely, and she's looking for something to rekindle her interest in life.
And that something just happens to be Milly Theale.
Susan's attachment to Milly might not be the healthiest thing in the world, but at least the woman isn't totally ignorant about it. She knows full well the purpose that Milly serves in her life, and even knows that "She had almost the sense of tracking her young friend as if at a given moment to pounce […] She struck herself as hovering like a spy" (126.96.36.199). She knows she's overbearing and possessive about Milly. She even realizes that Milly is the one thing in her life that she has over a rich old schoolmate like Maud Lowder: "Whatever Mrs. Lowder might have to show—and one hoped one did the presumptions all justice—she would have nothing like Milly Theale, who constituted the trophy producible by poor Susan" (188.8.131.52).
The fact that Susan thinks of Milly as a trophy begins to weigh on Milly as the novel unfolds, and it's little surprising that Milly tries to distance herself from Susan by telling her lies and taking Kate along to the doctor's office instead of Susan. But when Milly gets sick, Susan is always the one to stay by her, and in the end, she is without doubt Milly's most loyal friend.