Welcome to the Wefugees Community and thank you for reaching out to us.
Usually, the passport will be required to get married in Germany. However, as you also applied for asylum and you are in the middle of the procedure (what I understood from your explanations), you can also not ask the Jamaican Embassy in Germany to issue a passport. I would recommend you to get in contact with your local registry office and to explain the situation. At the end, this is the authoritiy which is deciding if you meet their requirements or not.
Good luck and please don't hesitate to get back to us with any further questions.
All the best,