I think she is acting appropriately for her age. That is not to say that she isn't working out some "issue." She is learning to express her emotions and deal with anxiety or frustration. I think it is important for her Mom and Dad to be calm and consistent with her. Prepare her for changes to her environment/schedule ("Grandma is going to give you your bath tonight, but Mommy will come tuck you in.") It is probably some of earliest signs of power struggles to come in the terrible twos. But the answer with a child so young is not to withdraw or force her into certain situations or punish. Instead, her parents should be even more responsive but also firm and consistent.
You, as an outsider, may want to look for a way to bond a little more with her. What are aunts for but to show up with ice cream and toys - are you above bribery? I am half kidding - you may want to work a little harder on distracting her when Mom is going and also spend some time with her and your sister so you don't become the person who is there "when Mommy goes away." She'll start crying the moment she lays eyes on you if she makes that association.
Understandibly, when Mom needs to leave, she needs to leave. The child has to accept and trust other people doing for her as well. At the same time, her Mom needs to continue to be responsive to her. We have all gone through this phase when our child feels super-clingy. It's normal. A child this age needs to feel secure. Only in this way will she learn to cope in stressful situations.
I think she is acting appropriately for her age. If anything, she may just be a very sensitive little girl who needs a little more coaxing or a little more cuddling than another child. I think it is unwise to label her as "spoiled." That implies that she is consiously being naughty. She is still very young, a baby, and I don't believe can be spoiled in the stereotypical sense. You cannot give a 15-month old too much attention, especially from her parent. A responsive Mom will lead her to feel secure and she will be a much more pleasent toddler and preschooler than a kid whose Mom tries to seperate herself from her child prematurely or tries to make her more "indepenent."
Remind your sister, that this too will pass. In a year, she will probablky be a daddy's girl and want nothing to do with mommy. So your sister might even try to enjoy all this extra "love." Remember, a secure child will be better equiped to soothe herself and deal with stress later on. So go ahead and "spoil" her now - it may make life easier in a year or so.