As I know so far, The ArcGIS Data Reviewer extension support ArcGIS Desktop Basic, I am wondering whether its functions or tools are different between ArcGIS Desktop Basic and ArcGIS Desktop Standard.
The easiest way to find out is to open Arctoolbox, expand the toolset and look to see if there is a 'lock' on any of the tools.
See http://desktop.arcgis.com/en/arcmap/latest/analyze/managing-tools-and-toolboxes/using-the-arctoolbox...
for specific details.
Normally, when you purchase an extension all tools are available and they should all be available regardless of license level
