Of the possessive is normally used for inanimate objects, so the bone of the dog is not a native expression. It means the same as the dog's bone , of course; these are both possessive forms, and normally mean a bone that the dog chews upon, though it is possible that it could refer to a bone inside the dog's body. The dog bone is not possessive, but dog is the noun used as an adjective describing the kind of bone: it may be a part of a dog's skeleton, or it may be a bone which the dog chews.
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