There is no area on the birth certificate for a surname for the child until 1st April 1969.
So for a birth in 1964, the child had no specific surname on the certificate. There was the mother's name, and the father's name (if present), and the child's first name(s).
When indexed, the surname was presumed to be the father's name if the parents were married, and the mother's name if no father was present. If the father's name was present but the parents were unmarried, both names were used for indexing. The child may have used either, neither, or both later in life, depending on the situation.