Welcome back to the Wefugees Community and thank you for reaching out to us again.
As far as I understand, the issue is not about the acknowledgement of the fatherhood (Vaterschaftsanerkennung), which usually takes place at the "Jugendamt". Your quoted text is about the birth certificate and the fact that the registry office (Standesamt) does not see your identities (as parents) sufficenly proven. In order to do so, they recommend you to apply for a "Amtshilfeüberprüfung", in which the German embassy in Ghana would most probably varify some documents and information about you/your identity in Ghana. As far as I understand, the Standesamt will give you a paper which says "Bescheinigung über die Zurückstellung der Beurkundung". If they don't issue this, you can ask for it. It might help in the meantime if authorities, the health insurance company or other instiutions ask for the birth certificate. This paper will however only show the name of the mother, not of the father. They mention the 28.8.2021 as deadline for presenting sufficient proofs of your identity. However, they also explain that you can do the "Amtshilfeüberprüfung" at a later stage (i.e. in a few years) as well. You do not necessarily have to indroduce the procedure now. In this case, they will issue a "Auszug aus dem Geburtenregister" (extract from the birth registry) after the mentioned date. It is not a "proper" birth certificate, but it is widely accepted by the German authorities as well.
If you haven't organised the Vaterschaftsanerkennung yet, you can ask for an appointment at the local "Jugendamt". If the child's parents are not married (or "only" married in Islamic law, for instance), the "Vaterschaftsanerkennung" is essential to include the father in the birth certificate or extract from the birth registry. At the appointment for the "Vaterschaftsanerkennung" you can decide if you want to declare the custody as well.
I will also link our dear @alla_fka and @mbeon-Christine here: Am I right with my explanations or is there anything you may want to add?
However, as the procedure is quite complicated and confusing, please also don't hesitate to get back to us if clarification is needed or you have any further questions.
All the best,