Harry is totally schizophrenic in MOR though. He's got all of the Founders in his head.

You seem to be working from a unified view of the mind in which there is one single personality with one single voice, and deviations from this structure are pathological. I don't think this is accurate.

Even if it was, it is common for people to hold internal dialogues, and not unusual for patterns to develop where certain kinds of thought are given certain labels. I don't think this says anything special about Harry, except that he has a rich and vibrant inner life.

Also, a Public Service Announcement: "schizophrenia" is an umbrella term for a long list of possible symptoms whose main common feature is disconnection from reality or warped perception of it. You are thinking of Dissociative Identity Disorder (commonly known as Multiple Personality Disorder), which is a completely different thing altogether.

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.