In some languages it would even sound very weird if you'd say that you are drinking soup (like Latvian- it is ALWAYS to eat soup). I'm not sure but I could guess that it comes historically because of the structure of soup. I suppose the spanish and chinese soups are more liquid- like a broth- with less vegetables and stuff. Latvian soups were historically much thicker, they were called "vira"- some kind of stew almost.
There is a third explanation; "to sup soup". Generally this means, to eat or drink in small portions by using the lips. "To sup" is a very old term, and can be traced throughout Europe. Nowadays it is used for a particular light evening meal, "supper".
I think it depends on the language, the culture and the dialect... I'm Flemish so I speak Dutch. In Dutch, it's no problem to say both "drinking soup" or "eating soup", but it still depends on were you live and your dialect. In my region we often uses "drinking soup". In English I'm not sure what to use, but to me "I'm drinking soup" sounds more natural. In Danish, I don't have the slightest idea, but I'd guess from this lesson they prefer "eating soup".
There is a difference between "Do the soldiers eat soup?" and "Will the soldiers eat soup?"The first questions if the soldiers eat soup generally. (Are they eating soup?) The second implies that they are about to make a choice either to eat, or not to eat the soup. "Will the soldiers eat soup?" would be "Vil soldaterne spise suppe?"
Edit: Correction as mentioned below: Your sentence "Will the soldiers eat soup" would translate as "Skal soldaterne spise suppe", and not "Spiser soldaterne suppe".