A picture of it taken with the camera at about 45 degrees to the surface that it is resting on in post #1 would give us an idea of the length with respect to the diameter. A picture of the other end would also be helpful. I think if it is a vibratory motor it would be atracted to a magnet. If it is an electret microphone I don't think it would be atracted to a magnet as all the ones I have seen have been made from aluminium. This is a link to some information on electret microphones that may help you to interface it with an auiliary audio input. If it is a vibratory motor it make no sense to connect it to an auxiliary input.
Les.
Given the single view and no hint about size or weight, it could be a lot of things. BUT NOT an electret microphone, because all of them tend to have the connections on the back.
Is somebody playing a game with us???
The first step would be to use an OHMMETER to see what the resistance is, and if it is affected by the meter polarity. An ECM will show a non-symetrical resistance, a piezo device will probably show an open circuit like a capacitor,and a motor will show a lower resistance. So there is a safe check that will provide a clue.
An additional thought is that it might possibly be a detonator for "plastic devices."
You don't need to speculate on what it is any longer. I did an image search. It is a vibrator motor for a Samsung phone. It will work with a 3.6V LI-Ion battery but I have no idea what the polarity should be.