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Exporting to shape file with database and/or feature attributes works very well in Bentley Map.
The core of Bentley Map is the XFM object model. This object model 'understands' attributes, which are stored as DGN resident XML fragments or inside an RDBMS (using the well-known MSLINK key on the MicroStation graphical element).
When using the Interoperability tools in Bentley Map, e.g. to create SHP or MapInfo TAB files, this object model is the starting point for exporting geometry and attributes. If the attributes are stored as XML fragments or as part of an RDBMS property setup, it should get export.
Besides the attribute storage used by objects in Bentley Map, the MicroStation platform offers more ways to store attributes: e.g. using tags or user data. These types of storage fall outside the XFM object model and will therefore not be exported automatically as attributes when you use Bentley Map's Interoperability tools.
To include attributes you will probably first have fit your data into the Bentley Map XFM object model.