Yes, I think what you're talking about is layers. You can have multiple layers, each with their own objects and properties.
GIMP can do this. Or Paint.net.
I know this feature requires an app that supports layers (at least internally). However I've tried paint.net and played with GIMP (a tiny bit), and both require you to manually create a new layer for each element you want to reselect and edit later. I guess I'd need something that will automatically create a new layer for each element.
MS Office has this as a standard feature for most objects (after inserting a picture, word-art, chart, etc. in a document you can select that object with a single click). It seems like most image editors would have a similar feature for inserted objects (i.e. text fields, shapes, etc).