Child support is often asked for, Either they cannot pay bills with the child living with them or are doing it so they can provide things just for needs, But yes she can. It may be out of spite or just the need of money to pay bills.

If she lived with her father for 13 years, the child support should have been sought from the mother. The evidence needs to be presented to the court that the father was well able to take care of the daughter in that amount of time. If anything this also suggests that the mother cannot support her daughter. If she could, she wouldn't ask and if she really wanted to, she would have been present during the 13 years your stepchild lived with her father. And as you say, not even a birthday card was sent. If that is the case than the father should ask the court to return the daughter back to him. What the courts will see and how they will act is on whats best for your stepdaughter, not who should get 'support'. It seems there is a bigger issue here.