Trophy child refers to a child used to impress other people and enhance the status of the parents. The child has to fit into the perfect picture of what the family is supposed to look like. They have to be in the best schools; they always have to look good. Their parents are so narcissistic, they can't see their child as an individual, only a reflection of themselves.
This sense of the phrase trophy child was coined by psychologist Lee Hausner in her 1990 book Children of Paradise. It's part of a long series of phrases that use the adjective trophy to mean "something used to impress others and enhance one's status."