Hi Martin,
> what the ArcGIS Data Store and specifically the relational data store have to do with the good old Enterprise Geodatabase based on RDBMS products such as Oracle, MS SQL Serverv or Postgres? Is this just a new name for the same thing, or is a relational data store truly a new relational database implementation?
Enterprise geodatabases (aka. previously known as ArcSDE geodatabases) are still relevant and part of the ArcGIS stack. They are the premier data structure to store and manage your GIS data based on a DBMS platform. With respect to ArcGIS Server, they are used as the "source" for data that powers web services - so they have been called "data stores" for ArcGIS Server. In this case, the term "data store" is a concept, which refers to registering your data source with ArcGIS Server.
Register your data with ArcGIS Server using Manager—Documentation | ArcGIS Enterprise
The ArcGIS Data Store was introduced in the 10.3 release and is a stand alone installation component of ArcGIS Enterprise. It is based on PostgreSQL, managed internally by the software, and is meant to be used as part of a "hosting server" deployment with Portal for ArcGIS. In a hosting server deployment, you would install: Portal for ArcGIS + a federated Server site + the ArcGIS Data Store, this enables you to have a Web GIS deployment in your own infrastructure. At 10.5, this is now referred to as a base ArcGIS Enterprise deployment, which enables you to leverage all the capabilities that a Web GIS offers.
Base ArcGIS Enterprise deployment—Portal for ArcGIS (10.5) | ArcGIS Enterprise
I've also attached some ppt slides from my Advanced ArcGIS Server UC tech session that discusses these concepts.
Hope this helps,