So by that logic nobody has any DNA in common with neanderthals, right? Try again. You’re missing a big, big, gotcha.
Of course people have DNA in common with their ancestors as a group, hell we share a lot in common with fish. But it's because these genes are common that they enter and re-enter your family tree at multiple points.
And your mitochondrial DNA definitely doesn't descend from Abraham, a male incapable of passing on mitochondrial DNA, so there's a reason I gloss over female line descent.
Good sign of someone arguing dishonestly is that when you call out their lies, they tell you that the truth doesn’t matter.