The setting is in a parallel realm
The reason why humans are absent in the setting despite real people and places being mentioned is because the characters live in a parallel world inhabited by all the mythical and magical beings humans have imagined - one similar to the Spirit Worlds of various world mythologies but with an Urban Fantasy twist (thanks to modernization). Dez's parents were the ones who gave Puffin sapience
Dez says that Puffin became intelligent after he wished for him to be able to talk and be his friend, but it's well established in the show that leprechauns can't grant their own wishes. HOWEVER, it is shown that leprechauns can grant the wishes of other leprechauns. Thus, it's likely that Dez must have wished from his own parents for Puffin to become intelligent. Also, Dez has mentioned being very lonely as a child, so it's not hard to imagine why he would make such a wish from his parents.