I have many different kinds of thread and 4 different machines. Fortunately each machine's bobbins are different enough to easily tell at a glance which machine they go with. I find that for me keeping the bobbin with the thread works best. In order to keep the thread from unwinding on the bobbin I used a tip from another member of the board and bought some clear tubing at my local Lowes and cut individual pieces that fit over the bobbin and keep the thread from unwinding.
My antique and vintage threads are kept in vintage thread boxes and the bobbins are kept in the bobbin areas in those boxes. Vintage threads are organized by material (cotton, or cotton/poly, or nylon, or silk) and color.
My modern threads are kept in a variety of tins and I use that Hugo's amazing tape stuff to affix the bobbin to the thread cone/spool. Modern threads are organized by material.
Rob