It should be possible to save a project to a file. Not the contents of all files but just the list of files and project properties. Then after deleting the project I could easily recreate it with import option. Or restore it on other machine.
Next step would be to automatically recreate a project (or open if exists) after opening project file with jedit (with double-clicking in platform's file explorer).
I have many many projects and most convenient way to handel them I can imagine is opening a project file which I keep in the project's folder. That's the approach of my previous editor.
Could you at least design such a feature? Then someone else could implement it.
FileSync doesn't do this. I think a good place for this might be the ProjectBuilder plugin.
If you look in the JEDIT_SETTINGS_DIR/plugins/projectviewer.ProjectPlugin, you will see pv.xml and under the "projects" folder, an xml file for each project. you can copy these over to another system.
If they are under an environment variable, and you are using plugin options - general - use variable-prefixed paths for storing project data, then those paths can work on other computers and even other OSs too.