Abby states she ran a DNA test her mother's hair in her locket. The hair was shown to be a clipping and would not contain the roots. The hair root is the only part that has DNA so Abby wouldn't have had a sample for comparison.
When discussing a mitochondrial DNA match, Abby says that there is a complete match, making the man who shares the match a full sibling.
Not only do they need not be full siblings, they need not even be half siblings.
Mitochondrial DNA is usually passed only from mother to child, with no male contribution; therefore any people with common female to female ancestry can have a complete mitochondrial match.
Abby and the man in question could be third cousins, with mothers who were second cousins, grandmothers who were cousins, great-grandmothers who were sisters, and a great-great-grandmother who is the common mitochondrial ancestor.
They could even technically be further removed than that.
Abby stated that she and her brother were "a total DNA match". That is not possible. Even if they were twins, they would be dizygotic ('non-identical'). Only monozygotic ('identical') twins (of the same biological sex) would be a total match. Only their mitochondrial DNA would match.
Abby stated that mitochondrial DNA was tested for the kidney donor search. However, only nuclear DNA contains the genes for the antigens relevant tissue-typing. Mitochondrial DNA would not be tested.
When Gibbs asks Abby what she's doing and Abby says "I'm searching birth records of Jefferson Parish", this is not possible. Louisiana is a closed record state. Those records are not available to the public.
It is stated that "the Cooler" has been hiding in suburbia, in a cul-de-sac in Woodly Park. Woodly Park is in DC, across from the National Zoo. No Washingtonian would call Woodly Park a suburb.