Does the exported shapefile show a shift to the original data set? If YES, you need to take care of the used coordinate system in eCognition!

The projection assigned to the export will be the same as the project projection which you can check under --> File --> Modify open project (Upper third depicts the project settings). For eCognition, the projections have to be in a projected coordinate system such as UTM and not in a geographic projection (lat., long.). You actually get notified if the coordinate system is geographic and you can re-project the data using --> Tools --> Reproject Data.