Welcome back to the Wefugees Community and thank you for reaching out to us again.
If we are talking about the birth certificate (which you actually don't sign in that sense), the civil servants at the Standesamt are always asking you to prove your identity and the naming history (right to bear the names you are giving them). From my experience, they want to see the parent's passports and especially their birth certficates in most of the cases. But at the end it is every civil servant's individual decision and every case might be assessed differently. If you can not prove your identity to their satisfaction, the Standesamt will issue a so-called "Auszug aus dem Geburtenregister" which most of the authorities in Germany also accept. Please see more information in this document. Once the procedure at the Standesamt is finished and you are not happy with the outcome, you may consider to appeal at the local family court (Familiengericht") in order to obtain an "real" birth certficate. Keep in mind that the court might ask you to pay fees - especially if you "lose" the case.
However, be aware that if you are holding a refugee passport (blue passport), the Standesamt should't ask you to provide or to apply for a national passport at "your" embassy and accept alternative documents as identity proofs instead. As a recognised refugee, people are supposed to not ask the embassies for any kind of services. This might endanger their status in Germany.
I hope this helps and please don't hesitate to get back to us with any further questions.
All the best,