Circumcision of the male only may have signified the special responsibility which God had assigned to the father. Both signify a union with God that has already occurred. But there are rather obvious differences which must be kept in mind. 170 Circumcision was performed on infants eight days old and evidenced the faith of the parents. Baptism was a public sign, circumcision was a private sign.
Baptism is for all believers, male and female, circumcision was only for the males.
Up to this time, God had promised Abraham a son but had not specifically identified the mother of this child. Abraham had been convinced by Sarai and circumstances that it must be Hagar. It seems as though Abraham still considered this to be the case. As for Sarai your wife, you shall not call her name Sarai, but Sarah shall be her name. And I will bless her, and indeed I will give you a son by her.