Attachment with the mother has a correlation with well-being, but in the case of extended families vs nuclear families, attachment is not always strongest with the mother. In cultures where extended families are more predominant than nuclear families, any family member may have a stronger attachment with the infant than the mother. This can impact the infant's well-being later in life since the traits developed in the infant are influenced by what personalitiy and traits the attachee has.
I agree that in different cultures other family members may have a stronger attachment with the infant, such as the father or the grandmother. However, this should result in the same effect as having a strong bond with the mother in nuclear families; as long as there is one person with a daily and signficant role in the infant's life, this should not affect the child's well-being later on in life.
