The 'round rubber thing" you spoke of might be an extra bobbin winder wheel. Or it maybe was replaced already and that is the old one.....

Correct! It is a tatting shuttle that is missing the circular bobbin holding thread, usually a fine ga., like 50 ga.. They are still available, but like everything else, way more expensive now days. It would look nice in an antique sewing basket though.

The bobbin is in there. It just isn't too visible in the picture.

That is a tatting shuttle for making hand tatted lace.

It doesn't go in a machine, it is held by hand and lace and tatting is made.

Yes, now we use plastic tatting shuttles; that was the old style metal shuttle. Makes lace decorations. (You work for hours to get a tiny little bit done!) Oh, you do this with your hands --- does not go in a sewing machine. They must have stored it in their sewing machine.