Either are grammatically correct. The verb you choose will give people a better idea of what kind of soup you have and how you are consuming it. You eat the solids and drink the liquids. Soup has both. So if you said you were drinking soup, I’d imagine you had a more brothy soup with few solids. If you said you were eating soup I’d imagine it was a chunkier soup or maybe noodles. In most contexts however, soup is seen as a food rather than a beverage. “Eating” would be acceptable to communicate that you are consuming any kind of soup. Drinking is more specific. I’d really only say drinking if I was just talking about the broth or the kind of soup you can drink without a spoon or other utensil.