Electronic – Do the very thin wires of headphones have a name

Yes, all of the conductive stuff inside the cable is copper. However, the copper wires are coated with a very thin insulating layer called enamel.

The enamel can sometimes be removed by rubbing molten solder on it.

If this doesn't work, then you might have some luck rubbing some fine emery paper on it.

As you can see, you don't need to clean your fingernails before you do this.

But one of the problems with headphone cable specifically is that it often contains some thread to reinforce it.

The thread makes soldering the cable pretty difficult. Usually, if I find threads in my cable, I just throw it away.

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.