Do German universities have "ethics boards"?
There is the notion of Ethikkommission (searching for it you will find many hits). 

However, this is in my understanding mainly concerned with ethical questions related to research activities, in the sense of question whether a particular research activity is legitimate or if there are ethical or moral concerns against it (rather than academic conduct or misconduct).  To interact with it  is quiet widespread and routine in the life sciences, I think.  In other disciplines it is by and large a non-subject. 
For the second quote you link to it could be relevant. 

As far as other subjects go  I am not sure if there is a uniform naming or legal regulations, but typically universities will have things like:


Datenschutzbeauftragte or a Datenschutzkommission for issues related to handling of personal data and alike.
Gleichstellungsbeauftragte or a Kommission fûr Chancengleicheit  for questions related to equal opportunities and discrimination. 
Behindertenbeauftragte for concerns of persons with special needs. 
Kommission zur Untersuchung von Vorwürfen wissenschaftlichen Fehlverhaltens for investigations of scientific misconduct. 


Instead of a Kommission it could also be an Ausschuss, or some other variation.  

At a university level such committees are often attached to the Senat, but they can exist on lower levels, too.