One of my youngest granddaughters says "Mine" a lot, even when things are not hers. My other grandchildren have noticed her behavior and have starting asking me, "Nanny, what do we say when she says that?"
Children around 2- or 3-years-old are just figuring out that they are actually small and helpless in a world that is not as simple as it seemed when they were infants. This is why toddlers are suddenly so bossy around people and so possessive of their belongings. Your granddaughter may feel a bit helpless among all of the other grandchildren and anxious to hang on to what is rightfully hers. Like all small children, she has a tendency to go overboard when she is worried.
Help the other grandchildren be especially kind to the smaller cousin at this time, but also point out reality ("Actually, this is my doll that I brought from home") and display empathy ("But let's play with the doll together today!"). Also, consult your grandchild’s mother and father to see if there may be other reasons behind the little girl's insistence that many things are "mine." Beyond this, I urge you to remember that your own children, like all children, once went through periods when they are just learning about the world.