Almost every soup recipe seems to call for sauteed onions. For instance, I'm looking up recipes for squash soup, and every single one calls for onions -- but every soup recipe I can think of uses onions, so it's not just this type.
So what's the purpose of adding onion?
Sauteed onions can provide both caramel flavors (from the sugars in the onions) and Maillard reaction compounds, depending on how they are sauteed. Thus onions can supply a range of "umami" flavors for soup which otherwise you need to get through roasting animal bones and other tissue (e.g. brown veal stock). Of course, even beef stocks often add onion as well for extra flavor.
As an extreme example of this, I often prepare a vegetarian French Onion soup using a meat-free broth made entirely from onions, onion skins, and cheese rind. Blind tasters often fail to distinguish it from a store-bought beef stock.
You can get a lot of the same flavors from combinations of other browned vegetables, but onions neatly supply a perfect package of flavor compounds in one inexpensive, long-keeping root vegetable. Why use anything else?
Onion is a great way to add moisture without adding liquid. It caramelizes beautifully and has a unique sugar. It also makes the palate water and it thereby stimulates a chemical breakdown and, likewise stimulates the taste receptor activity.
As a component of mire-poix, it is a traditional component of older soups (typically of European origin). The chemicals that make your eyes water open up your gustatory system to a broader spectrum of flavor.
It's a good flavor that goes with just about anything. I think I use onions in about 80% of savory dishes that I cook.
In addition to all the answers stating the flavour benefits of onion, it is also worth remembering that onions are packed with important nutrients which we don't always get enough of. Sulfur is important in a vast number of our physiological processes and onions and brassicas are excellent sources. They also provide dietary fibre, vitamin C, folic acid and a range of anti-oxidants and other chemicals and minerals which can aid the body in regulating everything from bone density to blood sugar levels.
Bear in mind also that soups have a lot of water, so you need to work at building up a flavour profile to stop them tasting watery, particularly if animal fats and proteins are in short supply. Soups and stews are often peasant dishes at root and onions are an easy thing to grow with access to only a small kitchen garden or allotment so for flavour and nutrient you get a lot of bang for your buck and onions and other alliums become traditional in soups.
Onions become traditional because they are extraordinarily good for us and while they may not be the most nutritionally dense package, being over 80% water, they have always been within the reach of all but the very poorest and also store well to provide vitamins and nutrients through the non-growing months.
Apparently we have been cultivating onions for so long that no-one now knows where the wild plants first originated, when european settlers arrived in the Americas they had taken onions with them to cultivate, but found that the americans were already growing and cooking them.
