If a child is born to people that are married to each other there is a presumption of paternity. It can be challenged but the presumption is the starting point. If the mother and alleged father are not married paternity must be established if child support, custody or a parenting schedule is requested from the court. Obviously, if there is no court involvement people are free to make their own arrangements.
For unmarried persons paternity is established with either a written acknowledgment of paternity or a court order determining paternity. The Court can be on consent of the mother and father or if not on consent then determined at a hearing which will most likely utilize a DNA test of the mother, alleged father and child. The lab will issue a report stating the likelihood of paternity based on the DNA samples provided. A birth certificate by itself is not an acknowledgment of paternity.
There are situations where a father has signed an acknowledgment of paternity believing that they are the biological father of the child only to later learn that they are not the father whether due to a blood or DNA test or even a statement of the mother. However, if the child is old enough to have formed a father/child relationship or believes the person is their father the court will not undo the paternity acknowledgment nor will they relieve the "father" of child support obligations. Likewise, the "father" will be permitted to seek custody and/or parenting time of the child. The reason is that the court is looking to protect the child from the emotional harm of having the person the child identified as the father removed from their life. This is true even if the mother tells the presumed father that he is not the "real" father and the biological father comes forward ready to accept paternity. Other than an adoption at that point the relationship with the child will not be changed.
Any man that has doubts as to whether or not he is the biological father of the child should challenge paternity as soon as possible to avoid being "estopped" (prevented) from having the tests and challenge made due to the age and relationship with the child. So if there is a newborn and the mother files for paternity and child support that is the time to have the tests done, not two years later because a friend of a friend told you that you are not the father.