Of course, you can serve soup in anything that will hold it. But over the centuries certain bowls have become traditional for certain soups, and for practical reasons. First-course soups are usually presented in smaller bowls than hearty main-course versions. Cream soups and consommés, which cool quickly, are traditionally served in comparatively deep bowls, looking literally like two-handled teacups with matching saucers; occasionally an elegant broth is served in actual demitasse cups and sipped like espresso.
The bowls of cream-soup spoons are deep and nearly rounded so that they can hold a reasonable quantity of liquid. Chowders, fish soups, and other soups with numerous ingredients, which are slower to cool, are usually served in shallow, broad-rimmed dishes or in underplates; the appropriate spoons for these are long-handled with shallow oval-shaped bowls that can hold plenty of chunks. If you can, serve Asian soups in Chinese or Japanese lacquer or ceramic bowls with those distinctive open-handled ceramic Asian soup spoons, and use ovenproof bowls or crocks for a soup such as French onion, which requires time in the oven. Any tablespoon will do, but soup is best eaten with a large spoon with an elongated bowl.
As for serving size, consider 1 cup of soup an appetizer, and 1 1/2 to 2 cups a main course.