I would suspect that there are not many 9 year olds that would be unable to confirm their name and confirm that the adult is either their parent , guardian or legal caregiver

The OP did not indicate that the child was his child and that he should have access to the booking

I would however have thought that the airline would have procedures in place to handle dealing with young children on bookings. I would have expected that the child's booking would at least have been cross referenced to the adult , so that it identified that the child would not be an unaccompanied minor and that the adult in the linked booking would be able to address issues

Children have just as much right to data protection as adults do and it is good to see that BA is attempting to do so - albeit in what may not be a perfect method