The drum of the washing machine is the part of the entire system that holds the clothes inside and spins them. The drum is made of stainless steel, with holes in it for draining/filling of water and cleaning fluids.
The shaft at the back of this part slots into the bearing which allows it to freely spin with minimal friction. The end of the shaft is where the flywheel goes on, which is in turn where the drive belt fits on, connecting it to the motor and spinning it. The end of the shaft has a specific shape (shown next to this paragraph) that fits perfectly into the flywheel so as to stop it from slipping when spinning.
The purpose of the shaft that can be found attached to the washing machine's drum is to hold the flywheel and connect the system to the motor so as to allow it to spin and cycle through the clothes inside.
Generally, this part of the washing machine is made from cast aluminum for its properties of being lightweight yet strong, able to resist oxidization, as well as being cheap, easy to manufacture, and as a bonus, easily recyclable.
The process used to manufacture this part of the machine is likely through turning, suggested by its circular shape. It is then attached to the metal piece that holds it to the drum likely via welding, and the three-pronged metal piece itself is likely made through die casting.
Aluminum is a lightweight yet comparatively strong material. This lightness is one of the reasons why aluminum is valued in these kinds of driving systems, as it reduces the total weight of the rotating components, allowing for more energy efficiency. Aluminum is also corrosion resistant and more importantly, resistant to oxidization. Aluminum reacts with oxygen to form a hard film of aluminum oxide that prevents further oxidization.
A video on the manufacturing process of aluminum can be seen below.
Because the drum is the part that both holds the load of clothes being washed and is repeatedly exposed to the cleaning chemicals, it must be resistant to chemicals (both cleaning fluids and rusting) as well as deformation. Hence, it is made of stainless steel, a very hard and chemically resistant material.
In order to manufacture this part of the machine, there are several processes it must go through. First, the curved surface is cut and pressed out of a flat sheet. The holes are then punched out, before the sheet is rolled and welded shut. The front and back pieces of metal are welded to the drum last, together with the shaft.
Stainless steel is a specific variant of the steel alloy, consisting of at least 11% chromium in order to protect against corrosion (rusting). Chromium works similarly to aluminum, in that it forms a thin layer of chromium oxide on the surface of the object so that the steel alloy under it is unable to react with oxygen and oxidize. Compared to aluminum, stainless steel is stronger, at the cost of increased weight. However, unlike aluminum, which softens at high temperatures and conducts electricity well, stainless steel does not.
A video on the manufacturing process of stainless steel can be seen below.