Welcome back to the Wefugees Community and thank you for your question.
I agree with @Nima, to register a car in Germany you need to provide an ID document (national ID or passport). If you submitted your documents to the immigration office, you may ask them for a certified copy and an explanation that they are holding the originals.
If it comes to the "Sozialamt" and the "Asylbewerberleistungen" (social benefits for asylum seekers) it is getting more complicated. In general, if you are receiving the allowance according to §3 of the "Asylbewerberleistungsgesetz" (you are in Germany less than 18 months) or even according to §1a (the authorities are sactioning you) any capital ("Vermögen") of more than 200 Euros will be considered in the calculation of the allowance. This changes as soon as you receive the allowance according to §2 of the respective law, the so-called "Analogleistungen". Now you will be allowed to have a "capital" of up to 5000 Euros as a single adult person. However, depending on the purpose for buying the car, for instance if you can prove that you are dependent on the car for job purposes, you may get an exceptional permission.
As you can see, it is not a question we can answer in a general way and it can get complicated. Maybe you can contact the Sozialamt directly or talk to a counselling office to get some support in this. It is important to be transparent, otherwise you might get in trouble with the authorities. They will probably get suspicious and want to know how you are able to finance a car with your social money. Would you agree dear @mbeon-Ruth?
For further explanations, you may also see this thread, in which we discussed a similar topic before.
I hope this helps and please don't hesitate to get back to us with any further questions.
All the best,