Exporting to shape file with database and/or feature attributes works very well in Bentley Map.
The core of Bentley Map is the XFM feature model. This feature 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 export tool in Bentley Map, 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, they can be exported.
Besides the attribute storage used by features 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 feature model and will therefore not be exported as attributes using Bentley Maps export tool.