The "some other stuff" in cakes, muffins, biscuits, pancakes, scones, etc. includes leavening agents, either baking powder, baking soda, or a combination of both. The "friendship cake" starter replaces that, just as the shoggoth replaces yeast in a yeast bread.
(I hope I explained that correctly. MG, anyone else, feel free to add/correct.
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Essentially correct. Most cakes use chemical leaveners (baking powder or soda). Most breads use yeast. A sourdough starter is a colony of wild yeasts, lactobacilli and other microbes (exactly what microbes are what give each strain of starter its characteristic flavor, and is why San Francisco sourdough tastes different to, say, a New England sourdough). An Amish friendship starter is a mild-flavored sourdough starter used to make a sweet quickbread with a crumb similar to a cake. It seems to have been started by a troop of Girl Scouts in the early 1990s.
