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Yes, this is perfectly true: this is covered in booking condition 5.3, so paperwork is needed when a parent has a different surname from their child ie: child from a previous marriage, or the parents aren't married so the child surname is different from one of the parents.
If this is the case, you'll need a papertrail to show the linking of names.
ie: mother remarried with child from previous marriage would need birth cert to show her old married name is mother to the child PLUS divorce cert to show old married name back to maiden name and marriage cert to show new married name from maiden name. I hope this makes sense.
I think this is a legal protection for the cruise line as chilldren with a different surname could be going abraod/crusing without the consent of the other parent, in other words you could be kidnapping them out of the country when perhaps and I know you are not you want to get them away from the father. A copy of their birth certificate and signed by the other parent [their father] might be the way around this but you need to be very careful this is correct, see what other contributors are saying.