Why does Day reach for his pendant? Why is it important to June? Why hide such a common item?

In Chapter 16, June and Day drink nectar wine in an alley. June asks what he is saving money for, but Day avoids the question and kisses her passionately. They admit that they’re attracted to each other. As the wine catches up with Day, he gets ready to go to bed and unconsciously grabs at his neck, where the pendant used to be. June sees this and remembers the pendant Day left at the hospital - which she carries in her pocket. She realizes who he is.