They just seem to go together perfectly without the need to add much more to make a tasty dish. However, for some reason, it doesn't seem to appear in traditional cuisine at all. Is pesto considered exclusively a pasta sauce? Is there a reason why?
I'm assuming you mean basil pesto, although according to an Italian dictionary a pesto is a sauce made by crushing ingredients (pestare is Italian for crushing). Some specific types (like the Genovese one) are protected by law and you can't call something with that denomination unless they have some specific requirements.
As Italian, I've seen basil pesto also as a sandwich sauce together with cold cuts and/or cheese, pizza topping or as lasagna filling as alternative to the ragù (aka Bolognese sauce outside Italy:) I think this answers your first question.
Together with meat, I've never seen it; probably because it can be too strong to cover the meat's flavor. However, a similar preparation with parsley (salsa verde) but probably not as strong usually accompanies the bollito (boiled meat). I'd say the pattern is that we're speaking about not-so-flavorful meats, but here I'm afraid to cross the line and step into speaking about my personal taste.
It simply depends on how much of a traditionalist you want to be. There are actually a few different types of pestos that originated in Italy, with the most famous being the Ligurian version that includes basil, garlic, pine nuts, olive oil and pecorino. As with many traditional foods in Italy, there is actually an organization (Consorzio del Pesto Genovese) that specifies the exact ingredients (and their source), and the precise process for constructing this version of pesto. Of course, these days, many Italians, and many people around the world use the ingredients and tools they have on hand to construct a pesto. While it was originally used, and is today most commonly used, to dress pasta, you can certainly find it in Italy, and worldwide, flavoring meat, pizza, breads, soups, and more. So, it may not be "traditional", but it does have other uses in Italian cuisine, and there is no reason you shouldn't continue to enjoy your pesto chicken.