Is the student obligated to return the discounted amount she received for her academic excellence to the entity that granted her the scholarship for her studies, or is this amount rightfully hers?

If the amount is a pure gift and a reward for excellence, then it is your right. However, if it is a discount from tuition fees and the granting entity will request its retrieval, then it is not permissible to take it without their permission, because a gift is restricted by the condition of the giver.