I've been offered a PhD position at a Max Planck Institute in Germany. My PhD needs to go through a German university to be valid. Do you think it would make any difference which university has formally issued my PhD? I can choose anyone I want, but would prefer one where I don't need a subject test in the end.

Uni does not matter. Max Planck institute is excellent.

I don't think it will matter. The Max Planck normally are loaded with money (unlike the universities) and very often have english as the lab language because they attract many foreigners. Which universities can you choose from?

My supervisor recommended Braunschweig.
The MPI is in Göttingen, but apparently you have to not only defend your thesis but also write a subject test there, so he adviced me not to choose that university.