Why is DIY yogurt not as stable as sourdough starter?
First, we are talking about different microorganisms when comparing sourdough to yogurt. Generally, a sourdough starter is populated by numerous strains of yeast and bacteria, while store bought yogurt contains a small number of isolated strains of bacteria.  However, there is no reason that you can't maintain a yogurt culture that lives on. According to the linked article, the isolated strains in store bought yogurt are generally only viable for a couple of generations.  You might look into heirloom starter cultures, as they suggest. That way you can develop a yogurt starter that is as viable as your sourdough starter.