It seems as the GIT page is the only source of information regarding system requirements (ArcGIS Desktop section) because at the Esri page, they refer to the github link.
ArcGIS Desktop 10, 10.1 or 10.2 is required for using the Geoportal
Clients for ArcGIS Desktop. The Geoportal Clients are offered in
versions specific to your ArcGIS environment; i.e., use the 10.1
Geoportal Clients with ArcGIS 10.1, and the 10 Geoportal Clients with
ArcGIS 10. The Geoportal Clients allow users to connect to the
geoportal from within ArcGIS Desktop to publish and search. Caution:
The ArcGIS Desktop .NET Framework option must be enabled during
installation of ArcGIS Desktop on any machine that will be using the
Geoportal Clients.
There is a good-looking Esri installation guide published as a white paper:
How to Set Up Esri® Geoportal Server 1.2.2 on Windows: Aggregate Geospatial Resources with Open Source Technology.
Esri has published two desktop clients 1.2.4 - one for 10.1 and another for 10.2, so this implies that geoportal is supported with ArcGIS Desktop 10.2 indeed.