I dont think its possible by configuration,
but below things you may look at,
1. Create custom property named "Projects"
2. Add this property in operation descriptor of the your custom objects, and make it mandatory and visible
so that it will be visible on item create dialouge.
3. Create custom LOV, which will have all projects in your organisation as values of this LOV.
4. Now one the project user has to select while creating the Item,
Now here you will require the customization for mapping this project value in Assigned project property.
Hope you get the idea.