What are the user.config files for in ArcMap and ArcGIS Pro?

Answer

In ArcMap, the user.config file is used for storing the registry keys and settings related to the search window for the user account accessing the application. The user.config file for ArcMap can be found in the following directory in Windows:

C:\Users\<USER>\AppData\Local\ESRI\DefaultDomain_Path_<STRONGNAME>\<VERSIONNUMBER>\

Note:
The Application Data (or AppData) folder is hidden by default. For more information on showing hidden folders in Windows, see Show hidden files.

In ArcGIS Pro, the user.config file stores personalized preferences such as geoprocessing, search, and mapping properties. The file also stores the location of recent projects accessed from the start page in ArcGIS Pro. The user.config file for ArcGIS Pro is found in the following directory in version 2.5 and below:

C:\Users\<USER>\AppData\Local\ESRI\ArcGISPro.exe_StrongName_<STRONGNAME>\<VERSIONNUMBER>

Note: 
At major releases past version 2.5, the user.config file is located in the Roaming folder:

C:\Users\<USER>\AppData\Roaming\ESRI\ArcGISPro.exe_StrongName_<STRONGNAME>\<VERSIONNUMBER>

The user.config files can be renamed, for example. user_OLD.config. Renaming the user.config files restores ArcMap or ArcGIS Pro to the default settings, but a new user.config file is created the next time ArcMap or ArcGIS Pro is launched.
The previous settings remain in the renamed file, and can be restored by renaming the file back to user.config and overwriting the newly generated user.config file.