I would like to know how is it possible to make a organizational Chart-Browser as shown in the image attached, in SharePoint 2013?
You can use Visio and Visio Services. Visio already has a org chart templates that can link to data sources and import pictures from exchange (if desired).
there are open API's available that can also be used. You can embed thse in SharePoint pages. for Example, Google has one here: https://developers.google.com/chart/interactive/docs/gallery/orgchart
There are also a few 3rd party tools that have webparts available.