Disclaimer: What you are about to read is all about my interpretation not the writers of the show and I'm not looking at his powers through a Doylist lens.
Nathan and his powers always only ever sort of fit for me but felt a bit off so I've been trying to think through some good links for his powers to who he is as a person.
Some have argued that Nathan's immortal because he's already unaffected by anything which I'd argue we know isn't true as we have plenty of examples of him being effected by things e.g him being kicked out by his Mum, him in the alternate timeline following the gang's deaths, anytime he's in the same room as his Dad, the entirety of series 2 episode 2 etc.
His Mum says that he's not effected by anything anyone says to him because she's bought into the version of him he likes to project to others as a sort of defensive mechanism that usually ends up fucking over other people. He tells Kelly that he doesn't get embarrassed by anything but we literally hear him thinking through Kelly in the very first episode "She can't throw me out. She's my Mum. I'm a homeless. That is so embarrassing." He then tries to hide his homelessness from the group until he has to reveal it in series 1 episode 3.
He likes to present himself as so outrageous and out there so that he can seem like he's unaffected by anything and that he has no filter (we know he does have one because of Kelly's power) because if everyone thinks he has nothing to hide or that he can't be hurt, they're less likely to go looking for his weaknesses/his secrets. Immortality is like the reinforcer of this.
He's immortal in the sense of resurrection but not in the sense of immunity to dying or any type of invincibility. He can be hurt and he can die. He'll just spring back afterwards but he's still vulnerable to hurt. Linking to how he'll be affected by things in his life (Ruth, his parents, Jamie) and then he'll try to move on like nothing's wrong but he keeps on getting hurt in the same way over and over.
An example is being left by people he cares about: his Dad leaves him when he's young, he tries to keep his Mum in his life out of a subconscious attempt to avoid her leaving/replacing him too but that backfires through the way he does it and she kicks him out (She says he can move back in later but he turns her down, evidently not wanting to stay where he thinks he'd be a burden with the knowledge that this relationship isn't stable in his mind anymore if it ever was), he's vulnerable to Ruth and then she dies, he opens up to/wants to care for Jamie and then he dies.
Something happens to him, he tries to move on and then it just keeps on being thrown in his face. Just like with how often he dies with his immortality. He'll come back but he's left vulnerable to the same thing happening again after.
I'd argue the mediumship/seeing ghosts is a symbolic way of showing that he is still affected by things as issues literally still continue to haunt him after they've happened. Though his ghosts do disappear pretty quickly afterwards but that can also be because there are only 6 episodes per series.