I would say it's HIGHLY possible "MOON" is about a generic "lost love", in which "lost" can be that person/thing has gone away, passed away or even disappeared.
I chose to see it as a song for her dog because in the PV we see a glimpse of her "past" in the same room, only she's happy and waving with her dog.
The initial writing might not be for it but I think along the way she got the idea the song could be dedicated to the deceased pet as well.
