Welcome back to the Wefugees Community and thank you for reaching out to us again.
First of all, the issuing of the birth certificate is not depending on the father. Instead, your friend and mother of the child may be mentioned as only "known" parent in the birth certificate or extract from the birth registry (please see this link for more information on the latter). If she wants, however, that the child's father is recognising the fatherhood and he is refusing to take responsibility, she can file a suit at the family court. The local "Jugendamt" can support her on this. I anyway recommend to contact the Jugendamt as they can reach out to the father and mediate between the parents and potentially make a case in court redundant.
If the child officially has a German parent, it can get the German cititzenship as well. As mother of a German child, your friend might be entitled to receive a residence permit according to section 28 of the German Residence Act. However, as she still holds an Aufenthaltsgestattung, her asylum case is still pending. At the latest when the father acknowledged the fatherhood, I recommend to contact a lawyer or a migration counselling office in order to discuss her individual situation in more detail.
I hope this helps and please don't hesitate to get back to us with any further questions.
All the best,