What is the Purpose of Using Onions in (Almost) Every Soup Recipe?
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?