I would just say that x was such a lovely baby that their parents chose them to come and live with them and be part of the family. Maybe follow it up with sometimes babies are born and their mummies/daddies can't look after them (because they might be too young or sick, etc) so you are very lucky x got to be part of your family.

With your cousin who was adopted, I'd say pretty much the same in reverse - cousin was such a lovely baby that the family she grew up with wanted him/her to live with them, etc.