If the man is married to her a child born during the marriage will be presumed to be the husbands unless proven otherwise in a court of law with paternity testing. If the man is not married, a paternity test will generally be ordered and always absolutely should be ordered anytime custody and /or child support become an issue. The first child's child support is taken into consideration when awarding child support to the second child. I have never seen anyone get in trouble for fraud for putting the wrong name on a birth certificate because paternity testing results are higher in the pecking order so to speak than the name on a birth certificate. It would seem like unless the biological father is a dead beat, drug addict, incarcerated convicted , has domestic violence charges on him, or someone like that, that the mother would want assistance in supporting the child. Children cost a lot of money to support. The biological father could petition the court himself for paternity testing if he thought he was the father and he could petition for visitation, support, and/or custody.
There are many factors that can affect how best to handle this matter and the best advise is to hire an attorney immediately. Unfortunately the advice above is a guess that is based on very scanty facts. Circumstances of all sorts can change the ultimate answer you need. If you want to know how best to handle the situation, make an appointment with an attorney and get good solid advise based on more exact facts. The money spent might give you the peace of mind you need.