Why does magnet wire need to be insulated?
It is true that it's only the volume and the power fed to the winding that matters for magnetic field, in electromagnets and motors. Therefore, you could have a single turn winding. 

Unfortunately, a single turn would (generally) require a very high current and a very low voltage. This is true on the scales we tend to work at, and the values that physical constants happen to have.

Practical electromagnets use a relatively cheap trick to increase the voltage and decrease the current, by splitting the short fat wire of a single turn into a long thin wire, wound round several times. As each turn has a different voltage, they need to be insulated from each other.

A huge advantage of thin wire in the winding is that connection wires can be a reasonable thickness, and still be much lower resistance than the working winding.

A disadvantage of this trick is that circular wire does not fill 100% of the available area, and the insulation consumes some space as well, so we lose some copper area compared with a single turn. However the trick is so cheap and useful that this inefficiency in area is a small price to pay for the benefits, for almost all applications (in some very big machines, square cross section wire or bar is used for windings to improve the packing density).