I think it would be depends on where you are from. In my country, its common to just say two soups instead of just saying two soups as to indicate different kind of soup. If you want to refer to specific soup, then it would be something like "quiero sopa de patatas/pollo" as like another examples given by duo.
"Two soup" is definitely never correct. "Soup" is generally considered to be a collective noun (it doesn't exactly have a plural, it's assumed to be coming out of a cosmic vat of soup or something haha), so it should usually be "two bowls OF soup" or "two portions OF soup" to reflect that. Sometimes you can substitute just "soup" for "a bowl of soup" or "a portion of soup," which means "two soups" works ok in that context. Never "two soup," though.