Exporting (Export > Export Selected Shapes) models to shp adds settings from the rules as attributes, while exporting to GDB adds only barebone information (start rule + rule file). Why?
What specific tool are you using and method? it should not do that if you do it the right way. I have worked with thousands of shape files and have never seen this.
OK where are you doing this?
In ArcMap under file there is no Export option. There is a Export Map option.
In ArcCatalog there is no export option under File
Use the Tool Feature Class to Geodatabase (multiple).
Exporting to GDB does not export rule attributes. You need to write reports in cga that report the values of the desired rule attributes in order to have this data in the GDB.