What makes this hot sauce last half a year in the fridge?
Nothing

Your instinct is right, there is nothing that would prevent botulism or other bacteria from growing at room temperature. This is a classic “but we’ve always done it that way, so it must be ok” case. Looking at the protagonists in the videos, I would assume that those are simply family recipes.

The problem is that there is a fundamental difference between safe food and not spoiled food. We can clearly define what makes food or a preparation method safe, but once something is not longer safe it doesn’t automatically become spoiled. 

In (traditional) cooking, unsafe methods are used often enough, for many reasons and sometimes on purpose, just think of steak tartare. And in many (I am tempted to say “most”) cases, with no ill effect. It’s often statistics that work in favor of the users. Consider botulism: The absolute risk of the bacteria being present and finding favorable conditions to grow is low, but the consequences of ingesting the toxin (not the bacteria) are dire. Refrigerator temperatures will slow down the growth, still I would assume 6-12 months too risky (considering that the official recommendation for garlic in oil is less than a week in the fridge). 

As always when getting recipes from random sources, use common sense.