I actually have a similar situation with my mother and niece. My mother has 2 birth certificates on file due to her mother remarrying. She has her original with her name and father matching. Her mother remarried and she had a new birth certificate issued with the stepfather's name, but they also used the stepfather as the birth father. She was officially adopted. My niece has one birth certificate with the original name and father matching. Her mother remarried, but the name only was changed to match the stepfather's. The certificate has an attached addendum note that is posted when the certificate is produced. She was not officially adopted by the new marriage, but she did have her name changed by court order.
So, it could be either situation in your case. Name change with adoption, or without. You may need to check with the agency that issues the birth certificates and use the mother's name and the birth father's name in one search, and then do a second search using the stepfather's name. If there was an adoption, the original should not be available or it will "flag" a note to the new certificate. Otherwise, you may need to check with the county courts and see if there are documents that may show the name change being processed. As for the dates being similar, but not quite exact, that could be intentional to mislead searching or just a mistake by the records keeper.