If the tool handles that get "moldy" are all wooden handles, then I'd be convinced it's because the tool handles have a coating of drying oil on them, most probably Tung Oil or linseed oil.

Natural oils from plants are a known food source for different kinds of molds, and bathrooms painted with the old linseed oil based paints were often the most heavily infested with mildew because the paint on the walls and ceilings provided a food source for the mildew.

Bar soaps are still made from real vegetable oils (notably Palm oil and Olive oil, from which the Palmolive Company got it's name), and the use of bar soap in showers results in palm and olive oil based soap getting all over the walls in a shower. It's that oil that forms a food source for mildew to grow, and so mildew is always going to do well in a shower stall where bar soaps are used instead of "Lipid free" skin cleansers like Aquanil and Cetaphil.

I don't know what could be causing mold to grow on plastic cords or plastic knobs (or plastic tool handles).

If wooden tool handles are turning dark on people, the cause is amost certainly because hard particles of dirt are being embedded in the relatively soft Tung or Linseed oil coating on the handle. I'd use a Magic Eraser to remove the dirt from the handle, and then give it another coat of a harder oil based coating, like MinWax Wipe-On polyurethane (which should be hard enough not to get dirt embedded in it).