Do I need a paternity test to get custody of my daughter, even though her father signed the birth certificate?

The mother automatically gets custody unless the father goes to court to get custody. Signing the birth certificate may not be enough to establish paternity in some states. An acknowledgement of paternity may be necessary.

If the father goes to court for custody or the mother for support, the court may order a paternity test to make sure of paternity.

Do I need a paternity test to get custody of my daughter, even though her father signed the birth certificate?

If you gave birth to the child then you are presumed to be the legal mother, is that your question? If not I don't understand your question.

Just as an aside, although it might seem obvious that moms giving birth to their children are legally the moms, occasionally that is not the case, such as in surrogacy arrangements (like where a mom only provides the womb services and carries another mother's child). I assume that's not what going on with you.