News article says humans, but earliest human (homo sapiens) was around 300kya. The actual paper uses the word hominids rather than humans.
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So the shared genes alone can't be used as a reason to call Neanderthals human.
Some human had Neanderthal ancestors, some did not, as you rightly suggest. However, all humans and Neanderthals share a common ancestor that's much more recent than our last common ancestor with chimps.
Ie we only have 'chimp DNA' in the sense that we share a lot of DNA with them, and that we haven't changed too much since our last common ancestor. Exactly the same is true of us and Neanderthals.