The dark side isn't even a personality, as such, which implies strongly that it's not a soul.

I was originally going to put a quote here, but it turned out to be pretty much half the chapter, so... Chapter 56. In particular, when you read

a blind terrified thing that only wanted to find a dark corner and hide and not have to think about it any more - [...]

Visualizing himself cradling his dark side like a frightened child in his arms.

Think back to Deathly Hallows, Chapter 36: King's Cross, specifically the bit

He was the only person there, except for -

He recoiled. He had spotted the thing that was making the noises. It had the form of a small, naked child, curled on the ground, its skin raw and rough, flayed-looking, and it lay shuddering under a seat where it had been left, unwanted, stuffed out of sight, struggling for breath.

He was afraid of it. Small and fragile and wounded though it was, he did not want to approach it. Nevertheless he drew slowly nearer, ready to jump back at any moment. Soon he stood near enough to touch it, yet he could not bring himself to do it. He felt like a coward. He ought to comfort it, but it repulsed him.