Is there a significance of the "Christian name"? The Unitarian Universalist church is living proof that there is some sort of naming component in the child dedication/baptism ceremony in traditional Christian churches. How is the Christian name assigned? By the priest or the child's family? What is the difference between the "Christian name" and the "baptismal name" and the "confirmation name"? And what is the significance of each name?
In some cultures, notably the UK but also others, the term 'Christian name' simply means (or used to mean) someone's first name. So David Cameron's Christian name is 'David'. The logic behind this is that babies were typically 'named' at their baptism, which typically occurred shortly after their birth. Going back thirty or forty years you would find official forms that referred to a person's 'Christian name' and 'surname', and the first name might be referred to like that even if the person had not actually been baptised. Nowadays the term has dropped out of favour as it becomes less and less likely that a person will be even nominally Christian. Someone who was baptized as an infant can still reasonably call their first name their 'Christian name'.

At Confirmation, Catholics and Anglicans may select a new 'Christian' name to go with their baptismal names, and adult converts are encouraged to do so. This is sometimes described as a 'Christian name'. It is not normally part of someone's official or legal name unless they explicitly change it.