"Bevo la mia zuppa."
"Zuppa" is a liquid or semi-liquid. It's not stew or chowder or even minestrone. Therefore, you drink a liquid. Few people, even in English, would try to say that we eat broth. "Eat" and "consume" have been equated in English, and they should not be.
Are you sure, Christopher? I did a quick Google search, and it seems you actually say "mangiare la zuppa". That doesn't surprise me at all, that's how you say it in Portuguese also. I think you're trying to apply English rules here.
But I would never say I was "drinking" soup. I think this is an incorrect translation. In English, we "eat" soup (though it doesn't "make sense") or, to stay closer to the Italian, we "sip" soup.