Why are copper cables round?

Like Jessica Rabbit- because they are drawn that way.

Wire production involves pulling (drawing) the wire through successively smaller dies (often with annealing in between). The dies are most easily made with round holes (they are typically made from very hard materials such as diamond).

Not all wire is round - rounded rectangular wire is sometimes used in inductors and transformers.

When you move away from circular cross sections you run into two significant problems:

First, corners would exert more pressure against adjacent things, and are more likely to result in insulation damage.

Second, for high frequency use the skin current effect would result in higher impedance in a cable with the same cross sectional area but with corners. Further, the greater amount of insulation required to cover that conductor would also change the impedance. While this could be dealt with, we already have a great understanding of the skin effect, impedance, and insulation effect at high frequencies for round conductors that it would be limiting for designers to use alternate configurations without a good reason (such as fitting more copper in a smaller area for efficiency in inductors).

There are many other considerations - wire drawing is already difficult and strenuous, having a circular cross section reduces die contact for a given area of copper. Insulation would increase due to the additional circumference of a triangle or square compared to a circle (again, for the same cross sectional area of copper). When bending or flexing the corners would see more stress, would harden more quickly, and would crack more readily than circular cable.

When discussing cables it becomes even harder to make other cross sections. Imagine taking 10 conductors to make a cable in a triangle. When bent, twisted, or squeezed you'd find the individual wires in the cable moving out of their place and deforming to a more circular shape. How would you keep those wires in place to maintain the desired cable cross section? You could go to a single wire, but then it will be much harder to manage, much less flexible, and more prone to breaking and damage.

Flexibility

A round cross section allows the wire to flex in any direction equally.