The obvious answer is that the structures are not identical. Biochemical reactions are determined by both composition and length of each chain. The analogy of "information" is the same as saying that if you throw a lit match at gasoline, the gasoline gets the information to ignite.
Regardless of the species which we are examining, the DNA structure will ALWAYS be the same double-helix configuration.
To discern which life form will be created by the double-helix structure in question, we examine the data contained in said DNA, stored via the sequence of A, C, G, and T bases.