When exporting models to separate files, the filename for each model will have the shape name appended to the base name. Therefore, you can write a python export script that changes the name of each shape to be whatever is in the object attribute. Then, you can specify the python export script in the last parameter in the export dialog window. To create a python export script, right click on the scripts folder -> New -> Python Module -> Module:Export. Then, add the following code (lines 4-7) to the initExport function which is executed before the export is performed. This code sets the name of the shape to whatever is in the object attribute called myName.
def initExport(exportContextOID):
ctx = ScriptExportModelSettings(exportContextOID)
shapes = ce.getObjectsFrom(ce.scene, ce.isShape)
for s in shapes:
ce.setName(s, ce.getAttribute(s, 'myName'))
Note: Setting the shape's name in the initModel() function will not set the file name suffix. In order for the file name suffix to be set, you need to set the shape in the initExport() function.