Only the natural parents can re-register a birth after marriage. This is a formal, legal procedure.
A baptism record is NOT the same as re-registering the birth. It records a baptism, that is all. A re-registered birth appears on the legal birth register. A baptism will not. So, if you searched the birth record for your Great grandfather I suspect that he only appears once on that.
The original birth record should ALWAYS show the natural birth mother. It would be very difficult to get this wrong. Sometimes the wrong father could be registered, for a variety of reasons. A re-registered birth also can only legally show the natural mother.
However, what may have happened here is that the natural mother died at or after the birth, but before the baptism. Or perhaps the natural parents separated. Then the father entered another relationship, and it is this that is shown on the baptism record. The baptism record is made by a church and has no legal weight. The church may not have been aware that the 'mother' was, in effect, adoptive, or may have known and not worried about it as they recognised the relationship. This could be down to whoever was in charge at the church at the time.
To repeat, ONLY natural parents can formally re-register a birth after marriage.