The little black cat doesn't have a name in the American remake but you can assume that it's the same as in "Ju-on" (2002). He is killed horribly by Takeo (Takashi Matsuyama) and comes back as a vengeful spirit.
Mar's owner, Toshio (Yuya Ozek), really loves cats. He even has a ceramic figure of one.
No wonder the ghostly Toshio even meows like a cat. You can see how he spent all that time locked in the closet. He was busy drawing scary cat pictures all over the wall!