Hello @sb and welcome to our community!
A religious marriage is not sufficient to be recognized in Germany. You need to have valid documents like a certificate of marriage that you only get through a formal wedding.
This has implications for the asylum procedure because you are not married to each other from a legal point of view. It basically means that both your cases will be treated as if they were not related to each other.
Also, there are a few rights that won’t be granted to you:
For example, if a married couple applies for asylum and one of the spouses receives an Aufenthaltsgestattung, then his/her partner won’t get deported till the procedure is finished. As a non-married couple you only have this right if you have a child together.
Furthermore the so-called “Familienasyl” doesn’t apply to you. It means that your family will receive a refugee status automatically if they come to Germany and if you were already recognized as a refugee. But you’d need to be married for that.
Besides, you have no right to family reunion if you are not married or related to each other. But that doesn’t seem to be that important right now because both of you are already here in Germany.
To answer your second question:
Marriage in Germany is possible for you if you can provide the required documents like valid passports. You have to get in contact with your competent Standesamt (registry office) and ask them what papers you need to hand in exactly. We had similar questions about this topic here in the community recently. Please check out the information in this thread: https://www.wefugees.de/k/questions/1997
I hope this answer was helpful to you.