In the case of Prussia, there is a rather original institution that was created specifically to handle (a small part) of its “inheritance”: the Prussian Cultural Heritage Foundation. Legally, it's a foundation created by a federal law, with funding from the federal state and the federated states.
Beyond that, property would not seem to be especially problematic. It could easily be divided between the newly created states based on a geographic basis. Also, the states that existed in the German Empire lost much of their power and significance during the Nazi era so that there would not be much left to inherit after WWII.
Instead, the allied powers created new states that correspond only roughly to earlier entities and got actively involved in their administration, at least for the first few years after the war. Even when the names refer back to historical entities, the borders and organization of the states therefore changed a lot.
For example, the modern state of Baden-Württemberg is not simply a merger of Baden and Württemberg but was created in 1957 from three different states that were only loosely connected to the pre-war entities of Württemberg, Baden and the Prussian “Hohenzollernsche Lande”. Even Bavaria, the state with the strongest claim to some form of continuity (the name Freistaat itself is intended to underline that) obviously got an entirely new constitution and saw many changes to its borders.
The regions that became part of the German Democratic Republic have a rather different history. The Soviet authorities did (re)create a few states and the German Democratic Republic maintained the fiction of a federal structure for some time but the states basically lost any significance and were finally simply abolished in 1958. They were recreated once again in 1990 to join the Federal Republic. For the most part, the East then took over the laws and structures of the West.
Incidentally, the merger between Mecklenburg-Schwerin and Mecklenburg-Strelitz dates from 1934. In fact, that's possibly the main turning point, not the end of WWII. Once the Nazis had reorganized and hollowed out the states of the German Empire and transferred their authority to the central government, formally ending them did not change much.
