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The student often has meltdowns because she is not as "smart" as my DD, or my DD has information she doesn't, or my DD made something she can't make, or my DD can sing better....the list goes on and on.
This sounds like a child whose identity is based upon being "best", therefore is highly competitive and highly insecure.

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it might appear that she simply idolizes my child
Jealousy?

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The school is aware of the situation and is trying to remedy the situation.
This could mean anything, from teaching the child that she has intrinsic self-worth and encouraging good sportsmanship win-or-lose, to attempting to contrive a situation in which this child triumphs over yours. Unfortunately, just as some parents unwittingly fuel sibling rivalry, some learning environments create unhealthy and counterproductive rivalry. Children often resent being a foil for others, rather than being recognized for their own unique strengths.

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My concern is how I help my daughter in the interim.
While some of this may depend upon the specific behaviors of the other child (and helping your child understand better behavior choices to make if/when she is in a similar situation) much of what may be helpful to your child may be reinforcing the same lessons which any child benefits from... ensuring she knows she has intrinsic worth and encouraging good sportsmanship win-or-lose. Competition can be brutal, especially for a non-competitive child. Stories of Olympic silver medalists and others who almost made it to the top and went on to other successes may help build resilience and perseverance to draw on later in life.

Being Me is a book about understanding, building, and maintain confidence. It is by Magination Press, American Psychological Association. It is not specifically about gifted kids, but does a nice job explaining personal confidence and social confidence in a way kids can understand. Parents may wish to read the book first, to see whether it is applicable in their own child's circumstances. Reading also helps a parent prepare to have meaningful conversations with the child on different aspects of the topic of confidence.

That said, a common trait in gifted children, often listed amongst identifying characteristics, which is alternately described as "advanced moral reasoning", "well developed sense of justice", "moral sensitivity", "advanced ability to think about such abstract ideas as justice and fairness", "empathy", "compassion". Links to lists of gifted characteristics include several articles on the Davidson Database here and here, SENG (Silverman), SENG (Lovecky). If your child exhibits extreme empathy and understanding, the other child may believe her own aggressive approach to your daughter is OK, if your daughter does not enforce healthy boundaries.