The problem with talking about souls is that there's no clear, universal definition for what one actually is. In a fantasy context, I guess it depends on what rules that particular universe has defined for them. This episode seemed to indicate that it's a species thing -- humans (and apparently angels) have souls, demons don't. Lucifer said so, and he should have a pretty definitive understanding of how souls work, given that he spent millennia in charge of their punishment. So that suggests a soul isn't something a demon can grow, any more than a human can grow wings. While a demon might care about people, that would just make her a demon who cares.
Of course, "Dad" works in mysterious ways, and could presumably choose to ensoul Maze if he wanted to, but would he want to? Or maybe souls are contagious? If you hang around humans too long and get emotionally/spiritually attached to them, you can "catch" a soul from them through some sort of spiritual osmosis? There are some who believe you put a bit of your soul into anything you create; I imagine it could also be believed that you share a bit of your soul with anything or anyone you love. Which might be where the concept of "soul mates" came from.
Or maybe Maze just needs to listen to enough James Brown and Aretha Franklin records?