As @maddy suggested and @bgh confirmed, this solves the issue. I just ran into the same problem.
Besides defining your custom Universal Type you can also hint that this type is a 'subtype' of a more general type. E.g. by explicitly specifying the UTI 'public.data' in the key UTTypeConformsTo as follows:
<dict>
<key>UTTypeConformsTo</key>
<array>
<string>public.data</string>
</array>
<key>UTTypeDescription</key>
<string>My App's File Type</string>
<key>UTTypeIdentifier</key>
<string>com.mycompany.myapp.sainputs</string>
<key>UTTypeTagSpecification</key>
<dict>
<key>public.filename-extension</key>
<string>sainputs</string>
<key>public.mime-type</key>
<string>steuerapp/inputs</string>
</dict>
</dict>
You can then specify your custom uti as an argument in the UIDocumentInteractionController:
UIDocumentInteractionController *documentController = ...;
documentController.UTI = @"com.mycompany.myapp.sainputs";