"Have" can always replace "eat" or "drink" in English. You have dinner, have a glass of wine, have a sandwich.
With "soup" in particular I would never use either of those verbs, as it's a liquid that is treated as food, so neither "drink" nor "eat" properly fits. As a native speaker, I'd say "have soup" 99% of the time.
And there's no confusion between this and possession. These are two different things:
- Have you any soup?
- Do you have soup often?