Welcome to the Wefugees Community and thank your for reaching out to us.
As far as I know, the authorities will still do the "Vaterschaftsanerkennung" but they might state in the document that the identity of the parent is not proven ("Indentität beruht auf eigenen Angaben" oder "Identität nicht nachgewiesen"). My colleague explained this also here regarding not submitting a passport in this context. If the identity of the parents (or one parent) is not clear, the child may receive a so-called "Auszug aus dem Geburtenregister" (we discussed this topic i.e. here) instead of a proper birth certificate until you both can submit the required documents. But this highly depends on the civil servants as they assess every case individually.
Do you agree with my explanations, dear @mbeon-fardeen? Or is there anything you would like to add?
All the best,