Small addition: the UrhG mentioned is the Urheberrechtsgesetz. Translated to English you would call it the "Law of rights of the originator". It is not just about "the right to make copies".
Urheberpersönlichkeitsrecht (what is called "moral rights" in English) is a subset of Urheberrecht.
UrhG covers copyright and moral rights, some stuff at the intersection, and then some, but on the other hand lacking a few bits covered by US copyright.
1:1 mappings between legal regimes seem to be quite rare.