In some of the chicken chili recipes I use, the recipe says to cook (or sauté rather) chopped onion with the chicken. Is that only because onion adds a certain taste? Or does this process do something more?
Onion tastes good, especially when caramelized, and since both it and the chicken go on to to be cooked for an extended time in the chili, there's really no food safety concern with cooking them together.
Both contribute to the finished flavor of the dish, and cooking them together saves time. There's no magical interaction between chicken and onion - if there were you'd see the same step in many different types of recipes.
Well considering that pan cooking is only one of many ways to prepare chicken, on top of not all pan cooked chicken has onions as part of this process, I am not really sure how to answer this in the way the question was asked.
Some recipes call for onion to impart flavor and tenderize the skin (I mean the outer layer, not skin proper) as its being cooked. For instance in chicken Marsala you don't want a tough seared outer layer, you want one that infused with the taste of lemon, onion and mushroom, but is just as tender as the interior of the nugget.