Since the soap goes all over the shower and not just on the caulk, why doesn't it grow on the tile or the grout? It does seem to get on parts of the metal door frame though.
To answer this part of the question, you need to know that there are two kinds of grout; epoxy grout and cement based grouts. Epoxy grouts are impermeable plastics and never need to be sealed to prevent mildew from growing on them. Cement based grouts (both sanded and unsanded) dry porous and need to be sealed to prevent mildew growing on them. So, if you have epoxy grout, you'll never need to worry about mildew growing on your grout, and so the rest of this post will confine itself to cement based grouts.
In a nutshell, the reason why there's no mildew growing on your grout yet is because it's still too alkaline, and therefore not to the mildew fungii's liking. But that will change with time as described below:
There is a cycle in nature called the "Lime Cycle" and it's important for every DIY'er to know and understand that cycle and how it affects masonary products, like concrete, concrete blocks, brick mortar and cement based ceramic tile grouts.
Here is the "Lime Cycle":
Marble is nothing more than limestone that's been compressed by the weight of the Earth's crust or oceans and heated by geothermal heat. So, when Rome fell, Roman citizens pulled the marble off the walls of the Colloseum and other public buildings and burned it in fires to make a white powder called "quick lime" or "calcium oxide" (formula: CaO). When you mix quick lime with water, you get a slurry called either "slaked lime", and if you let that slurry dry to a powder you have "hydrated lime" or "calcium hydroxide" formula: HO-Ca-OH, which is the "lime" we mix with portland cement and aggregate to make concrete mix in bags. We also mix that lime with portland cement and sand to make brick mortar. We mix that lime with white cement and sand (or not) to make unsanded and sanded ceramic tiling grouts and we mix that lime with Plaster of Paris to make real lime based plasters for the interior walls in older homes.
But, take a look at the formula for hydrated lime powder: It's HO-Ca-OH, and it's those hydroxyl groups (-OH) that makes the lime highly alkaline. So, when you mix hydrated lime with porland cement, sand and/or Plaster of Paris (which is gypsum), the slurry you get is highly alkaline, and as it solidifies it remains alkaline... for anywhere from about a year to two years.
During that first two years the alkalinity of fresh concrete, brick mortar, lime plaster and ceramic tile grout will diminish as the CO2 in the air reacts with the hydrated lime to form calcium carbonate, which is the principle constituent of limestone, like this:
HO-Ca-OH + CO2 makes CaCO3 plus an H2O molecule (which turns into a butter fly and flies away).
So, the lime added to your ceramic tile grout when it was manufactured is gradually turning into limestone, and as it does those hydroxyl (-OH) groups in the grout disappear and the alkalinity of the grout subsides.
How that affects you as a DIY'er is that you have to be careful painting fresh concrete or concrete blocks because if you use an oil based paint, the fatty acids in the oil will react with the high alkalinity of the concrete to convert the oil based paint into a crude form of soap, and the result will be that the paint will disintegrate on the fresh concrete, concrete blocks or lime based plaster. That reaction is called "saponification", and all soaps are made by using that saponification reaction to convert animal or vegetable fatty acids into soap in the presence of a strong alkali.
aside (If you have to paint fresh concrete, you can use a special acrylic primer made for this purpose on it after only a month or two, and then paint over that primer with any paint you want. The primer effectively acts as a physical barrier between the highly alkaline concrete and the top coat of paint. Once concrete, concrete blocks, brick mortar and the like are more than 3 or 4 years old, their alkalinity will have subsided sufficiently that you can paint them with anything without concern about alkalinity being an issue.)
In the case of your bathroom walls, THE ONLY reason mildew isn't already growing on your grout is that it's still too alkaline for mildew's liking. As the alkalinity of your grout diminishes, you'll find more and more mildew starting to grow on it, and in time you could be looking at something like this:
Mildew won't grow on the surface of wall tiles because ceramic wall tiles are glazed, and so the smooth glazed surface of the tiles prevents the mildew spores from grabbing onto anything to anchor themselves in place. Basically, they get water cannoned off the glazed wall tiles by the water spray and end up going down your shower drain. Porous grout, on the other hand, is a mildew spore's idea of heaven. It has a strong porous surface that mildew fungii can root itself into well, and a continuous supply of food every morning and many friday and saturday evenings.
I have a steel sink in my kitchen with clear caulking where it meets the granite counter top. Some of that same black discoloration is in the clear caulk and I've rarely, if ever got any soap in these areas. Any ideas here? Will the bleach/borax work on this too?
Bar soaps are most commonly made by combining vegetable oils with sodium hydroxide (formula: NaOH), also called "oven cleaner". If you use potassium hydroxide instead of sodium hydroxide, you tend to get liquid soaps like skin cleansers. To mildew, it doesn't matter what kind of soap it is, as long as it's got fatty acids in it, it's food. It's very possible that you have some liquid soaps on your bathroom vanity that may have gotten onto that caulk.
My experience has been that the less time mildew has grown in an area, the easier it is to remove with a borax bleach slurry.
I very much doubt that bleach would harm your granite. Bleach works by releasing lone oxygen atoms, and those lone oxygen atoms react with anything unstable enough to react with them, and those tend to be things that would decompose on their own given time, such as large organic molecules. The stuff in granite won't decompose on it's own no matter how much time you give it, so I expect any kind of natural stone is too stable to be affected by bleach. I'd say that the bleach/borax slurry would work equally well on the clear caulk around your sink, too. If it turns out that it doesn't, it's not hard to remove caulk and replace it. It's just that with silicone caulk, you need to remove it completely before the new silicone will stick.
Are there any bar soaps that don't have the fatty acids?
I would say that every bar soap will be made of vegetable oils (and in rare cases, animal fats) and so every bar soap will contain fatty acids.
The difference between a detergent and a soap is that a detergent is a synthetic soap. It's made out of chemicals, but it's works the same way as soap to emulsifying soils in water. The advantage to detergents is that they can be formulated to avoid a lot of problems inherent in soaps. For example, the reason you never see a soap scum ring in your KITCHEN sink is because you don't use bar soaps in your kitchen sink. You use dish washing detergents in the kitchen sink that can be formulated to either not react with the hardness ions in water, or to not lose their solubility in water even if they do react with the hardness ions.
So, if environmentalists use Dawn dish washing detergent to clean the crude oil off ducks and pelicans after an oil spill, I'd say it'd be worth a try using Dawn dish washing detergent in your shower to at least avoid getting soap scum all over your walls, and hopefully deny food to the mildew spores in your shower.
I don't know whether there's any vegetable oils in Dawn, so I'd trust a skin cleanser that advertised that it was lipid free to deny mildew a source of food before I'd trust a dish washing detergent to do that.
PS:
You can learn an awful lot about the chemistry of cleaning at the American Cleaning Institute's web site (formerly called the American Soap and Detergent Manufacturer's Association) at:
http://www.cleaninginstitute.org/