This sentence (Das Kind mag seine Mutter) implies the sex of that particular child, doesn't refer to the gender of the german word "das Kind",
The form for "the child likes her mother" would be "Das Kind mag ihre Mutter" - both ihre and seine are in the feminine as they proceed Mutter but specify the sex of the child.
No, sein is the possessive for his AND its, masculine and neuter. Das Kind is neuter so uses sein as the possessive. Die Mutter is feminine and so we must add an -e to sein hence seine. The reality is that Das Kind mag seine Mutter should not accept The child likes his mother as a valid translation.