When making fermented food at home, particularly those that are made without the addition of a starter culture, how does one ensure that the correct types of bacteria are carrying out the fermentation? Is this generally due to the properties of the food that is fermented?
A "starter culture" is just a culture of preferred organisms. They probably differ little from local wild organisms. For example, a mixture of flour and water left to stand will (with luck) be colonized by yeast in the air that result in a serviceable yeast culture. Control of this process entails nothing more than observation over time. The environment and nature of the medium suffice to control the bugs that colonize it.
Both the Korean and Chinese versions of kimchee depend on moisture and acidity (and perhaps salinity) to inhibit unwanted molds/bacteria. The process is evidently self-controlled, but I recall seeing a film in which it was said that failure to add salt to the mixture could result in spoilage. Note that in the linked article, the author says cavalierly that one can scrape any mold that forms off the cabbage and continue with the fermentation because mold will not grow in liquid--patently false but probably "true enough" not to affect results.
Aspergillus oryzae colonizes rice, and rice with Aspergillus growth can be subjected to a secondary yeast fermentation that yields sake. The commercial production uses highly-prized starter cultures for both the mold and yeast, but it is possible to substitute other varieties of Aspergillus. For example, one can induce A. niger to grow on a citrus peel, and transfer the colony to rice. The sake that results is fine. But in this case one would want to rule out species of Aspergillus that are associated with toxins. For similar reasons some care is taken in sterilizing containers used to make wine.
In any of these processes there is a possibility of colonization by the wrong flora. The cultures are controlled by the content of the nutrients and climate and observed carefully. Over centuries there have probably been many unfortunate accidents involving unwanted organisms. There is nothing like violent illness to bring the integrity of a process into focus.
In short, the control of home fermentations lies in (1) statistical likelihood that the right organism is present in the environment; (2) the nature of the nutrient and climate; (3) the experience of the person(s) monitoring the process; and (4) luck.