You exist on the same planet as everything else. There is a lot of overlap in the forces and chemicals you experience. You have lots of proteins, including enzymes, which to help you survive on that planet, so it makes sense that other organisms will have the same/similar proteins to some extent. These similar proteins will be coded for by similar DNA. If it was too different then the protein may not work and the organism would possibly die.
As such, you may share the similar DNA to other organisms. The closer you are (generally) the closer the similarity should be.
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