With regards chromed discs on bikes, they look great but are less efficient in the wet, and as you said, once the chrome starts to go...............
The brake discs on my bike look like stainless steel.
To be fair the discs on my motorbike look like stainless as well, but the rear wheel has a disc almost the size rear one on the Jazz, and the front wheel has two large ones as well - so roughly 2/3 the braking area of a Jazz on something that weighs less than 200Kg - but with less tyre area in contact with the road ( bike also has ABS ) - so friction not so important. Because of limited space for discs on a small car that has maybe 14" wheels as an option they have to use plain steel or cast iron to get better friction ( and a slightly rusty surface has more friction than a clean one ).
Car brake discs are all in cast iron (low carbon percentage), apart from some supercars with carbon-carbon or carbon-ceramic composite. Some discs for karting and motorbikes are made in steel, but it's not frequent. Steel is lighter than cast iron, but it works at higher temperatures so you have to increase pressure to the lever and pads last less kilometers.
Bike discs are different in shape and structure from car discs. They are drilled, both for better cooling but also to increase friction and so to remove the hard skin on pads after kilometers of very gentle usage. A bike is light, so you don't need a huge pressure on the lever, so it can happen that a very hard skin like glass creates on pads surface, with a low friction coefficient: holes on the disc help to avoid this skin creation.
Car discs are very thicker, the pressure sent to pads is much higher considering servo and the pedal lever. So discs must be more robust (and cast iron is more robust than steel at high temperature), and can be casted with a internal duct to vent and cool them.
The rust on the friction surface is standard in wet climate and does not affects functionality. After the first brake pressure the surface becomes clean again.