Coffee to Coughan? Could the person have spoken with a brogue & been misunderstood? Was Coffee a woman? I ask because a mistake on a woman's record in the pre-right to vote days could be taken lightly by authorities. Once she married it was her husband's name that needed to be correct. Yes, some modern birth records would ask for mother's maiden name - midwives & doctors could pass on incorrect info. Correcting the record would have involved getting all the parties & witnesses to town hall & someone would say why all the fuss!
If the woman had not been born in USA & the govt wanted proof of birth & lineage then a baptismal certificate might be the first & only document with her name on it prior to her marriage in the 18th, 19th & early 20th centuries.