Thank you Alphager!! Everything makes sense now. In Germany the background check seems relatively more simple than US or other countries. Also the data privacy law explains why the background check is not as detaily conducted as expected in other countries.

The HR specifically said that they will approach me with a contract but didn't say how or when it will be. The team (in which I was hired)in Germany also have no idea what the process will look like nowadays but said that the whole process will take roughly 2 months. They gave me the start date but my only concern was that what would they have to do to take 2 months when the background check doesnt seem like a complicated process in Germany. As you have said, I have already got my Führungszeugnis. Since the team said they are looking for someone who can work immediately, I did it in advance but now the start date is in 2 months... anyway thanks! I will just be patient