I think it may be something like FFC/FPC cable, which is often used nowadays for PCB connectivity.
It's not the easiest stuff to hack being very delicate. Some is made a bit tougher and intended for e.g. hot bar soldering but I think not the type shown in the photo.
Is it flexible?
You may be able to make a connector/switch from a couple of bits pf PCB, depending on the tools available to you. I have etched an connector a few times for soldering display FPC connectors to, but it's not the easiest and this is a bit different anyway as mentioned above. You want something that clamps onto the contacts.
For more of the cable, look on Digikey/Farnell/Mouser/RS etc, for FPC and FFC cable (and connectors, and similar speakers (look at datasheets and they will usually say what connector to use). You may find something you can hack into what you need, or maybe even some ready rolled switch with the required connections (though I am not aware of any, will update if I find something)
I do think this will be very difficult to accomplish, but it's always fun to try :-)
The pitch (space between contacts) looks pretty small - what is it roughly? (hold a ruler with mm markings beside to judge)
Best Answer
Headphone wires are so thin, because the power transmitted is low, and there are technical and economic reasons to keep the wires thin.
Proper speakers use large amounts of power to drive the magnetics inside them, to get a good decibel (sound power) output.
Little earbuds go straight into your ear canal, the transmission power can be extremely low in comparison, and it's very directional. The tiny little speakers in those need such low power, that thin wires is fine to transmit on.
Remember the more current you need through a conductor, the larger it should be to avoid losses due to good old Ohm's law (voltage dropped through a resistive conductor, due to current). As sweber mentions in the comments, the length of the wire plays a part in how thick the wires need to be - if the cable is long, like in a speaker system, the resistance needs to be kept low over the distance, and thicker wires (cross sectional area for a conductor) help with this. Since earbud cables are quite short, usually about a metre long, and for the expected current draw, thin wires are acceptable.
Finally, there is an economic benefit to not over-sizing the conductor for earbuds. Copper is expensive, especially if you are manufacturing millions of units of something, there can be a huge difference in price between one gauge wire and another, from the weight of copper involved. If you can technically transmit on a thin wire, and it's economic to do so, then why not? They are usually stranded wires to help with flexibility and reduce metal fatigue, and the wires are almost always coated with protective plastics or fabrics to reduce stress on the wires. The rubberized insulating around the wires is to avoid axial and shear stress from breaking the wires.
So no, it's not a conspiracy to make you buy more crap.