The sentence contains a dangling modifier. The phrase "A thirsty animal" suggests that the puppy is the subject of that description, but it is positioned in a way that makes it seem like it could refer to something else initially, potentially leading to confusion.
To clarify, you could rephrase it as: "Becky Blackbell's puppy, a thirsty animal, drank my entire pot of soup." This structure makes it clear that "a thirsty animal" refers specifically to Becky Blackbell's puppy.