–The Owner, displaying the personality that so endears her to the fan-base
The designation Owner Of "James" is used here for an otherwise unnamed character.
In the series
This woman claimed that Mayu's dog, Wanta, was her dog, James. Seeing the pair on the beach, she grabbed the dog from Mayu, accused her of stealing him (in the manga), and contemptuously dismissed Mayu's request to visit him sometimes. At an unknown point very shortly after that, Wanta left this woman and went to Maple House, just as Mayu agreed to live there. Despite Mayu's worries, she has not seen again.
Manga panels before their confrontation indicate that this woman was possibly following Mayu. She confronted her only when the two were alone, leaving room for possible doubt as to the truth of her claims of ownership. Another possibility is that Wanta had bonded so thoroughly with Mayu, this woman no longer mattered to him. Wanta did not bite or struggle against her when she seized him. An even simpler possibility is that Wanta chose to leave someone who saw him as a possession in favor of someone who was his friend.
Why this woman never reappeared to challenge or chastise Mayu once again is not known. A common fan theory is that she was among the many killed by Lucy. A slight resemblance between this woman and a victim of the re-emergent Lucy after she escapes and kills Professor Kakuzawa exists, but after all, these were from the same mangaka. If Lucy did kill her, it's likely this was not done on Mayu's behalf, as it was only much later in the manga series that Lucy showed any real regard for anyone at Maple House besides Kouta.