In English it's more normal to ask if a soup has meat. So I would phrase this as "Does this soup have meat?". If you say that a soup or a stew is "with meat", it sounds more like the meat is served on the side.
Does this soup include meat, or variations. The argument about "with" meat is the literal translation and would be acceptable in UK English if you were presented with a list of soups and wanted to know if a particular soup included meat, which also is not accepted.