"Officer Brady shook Ralph's paw and scratched his ears."
It's 100% definitely a proper noun, so your book is wrong, I'm afraid. The fact that it has an apostrophe to indicate possession has no bearing on things. And nor does the fact that it's the name of an animal (presumably a pet). In this case "Ralph's" is a proper noun in the 'genitive case' (genitive is just a posh word for possessive!). Here are some more genitives:
The dog's tail / The cat's paws / The lion's claws (Common nouns)