How would we know a particular star belongs to Andromeda rather than the Milky way if, as the article suggests, there is already some overlap between the two galaxies?

IANAA, but I think they're handwaving a bit about ownership, and the real point of the discovery is that there are stars in a place that we'd previously thought there would be a gap between galaxies. The money quote is:

"“It challenges the notion that galaxies are islands with big, empty space between them,” GuhaThakurta told Astronomy. “There's this whole idea of Immanuel Kant — island universes was what he called galaxies. It's challenging that idea.”

“I won’t be surprised if there are a few dozens of stars which are being contested by the two galaxies,” Feng added."