"Dog's" is ambiguous in itself (dog is / that belongs to the dog), but this doesn't cause problems in practice. It's exactly the same ambiguity as spelling "it's" for the possessive would give. Also, it's / its is only unambiguous in writing. In speech, they are identical, in every accent of English - and yet people understand each other perfectly fine in spoken English, so the ambiguity is not a problem in practice.
