Usually the "flavorless" version of various desserts or drinks is in fact vanilla flavored -- why not leaving the product just as it is, especially when the "flavoured" versions usually don't contain vanilla? Any why vanilla, not anything else?
Vanilla is a rich and powerful flavor, with over 200 flavor compounds besides the basic vanillin. Due to its complexity and neutral flavor, it can be combined with a plethora of other flavors without conflicting. It just provides a good neutral base to build on. The vanilla also enriches the products if they are combined with additional flavorings, adding body and depth to the final result.
Ok, so Vanilla makes sense, but why add ANY flavoring?
On their own, sweet products may not have much flavor (remember, ice cream is just milk, sugar, and cream). Vanilla gives them a little flavor as a starting point. It also masks any unpleasant flavors present as a result of ingredients. For example, truly unflavored yogurt has a fairly strong lactic acid flavor of its own; this is something where you can get a TRULY unflavored yogurt and see what I mean. It's a whole different animal, and brings to mind that it's simply soured milk. Carbonated water similarly has a fairly unpleasant bite to it when no flavoring is added, as a result of the carbon dioxide reacting with water to produce carbonic acid.
Why can't I find stuff without ANY flavoring?
You can, if you know where to look! The truly unflavored products are generally used differently than their vanilla-flavored "unflavored" counterparts. Club soda / seltzer water is simply unflavored soda, and is stored with the stuff for mixed drinks, since that's where it is normally used. Unflavored yogurt is used in cooking savory Mediterranean dishes, and may be to the side of the flavored yogurts. As the comment mentions, unflavored ice cream is available as sweet cream.
I can't think of any other normally-vanilla flavored products, but if you can come up with some, I can probably find them for you under a different name.