"And, drum roll please, if these tests worked, they should still give the same answer for any person whose parents are the same. The fact that the article confirmed through a proper DNA test that they were identical notwithstanding, one daughter cannot be 13% from somewhere and the other 3% from the same place, if they have the same parents."
While yes, the article confirmed the sisters actually are identical twins, what you stated is incorrect.
In the case of non-identical siblings, assuming meiosis completes correctly, each sibling inherits approximately half of their nuclear DNA from each parent, ignoring the discrepancy between the X and Y chromosomes. Since these functional halves are not identical between siblings, each sibling receives a different percentage of ancestry from any particular source, only guaranteed to inherit specific sets of genes in common. For instance, any two brothers will inherit practically the same Y chromosome, and any two sisters will inherit practically the same X chromosome from their father. As such, in the case of two non-identical sisters from the same parents, one daughter actually can be 13% from somewhere and the other only be 3% from the same place.
Taken to the extreme, though highly improbable, it is theoretically possible for two siblings (one genetically male and one genetically female) to be born sharing no common nuclear DNA whatsoever outside of what is common to our species, inheriting the exact opposite halves of each parent. In this scenario, it would be possible for one to have no common ancestry to the other in their nuclear DNA. The chances of this happening are so infinitesimally slim that we can reasonably assume that this never has happened and never will happen in the entirety of our species, but it is possible.