Project status bar is enabled by Projects feature - it's enabled/disabled site-wide and selecting project while editing the page adds it to the project.
Project gadget is enabled on per-user level and unlike project feature it does not auto-add edited page to selected project and requires dragging item to the project.
There is a good explanation of difference between these 2 here: https://support.optimizely.com/hc/en-us/articles/4413200671501-Manage-projects

I just created a test Alloy site on CMS 13 and looks like the difference is that in CMS 13 Projects feature is disabled by default.

There is contradicting documentation about this:
https://docs.developers.optimizely.com/content-management-system/v13.0.0-CMS/docs/projects - says The Projects feature is turned on (default) or off for the entire site and affects all users
but https://docs.developers.optimizely.com/content-management-system/v13.0.0-CMS/docs/cmsui-section says 

And it looks like the latter documentation is right because after I set ProjectModeEnabled to true in appsettings.json and restarted the app - Project bar appeared.
Just in case - this is what I added to appsettings.json

  "EPiServer": {
    "CmsUI": {
      "ProjectUIOptions": {
        "ProjectModeEnabled": false
      }
    }
  }