I think it's odd he's describing it as "psychosis" and describes it as bordering on musical hallucinations. Like he's trying to make it sound more serious and important by borrowing medical jargon. It doesn't need to be described in those terms to be taken seriously, if it bothers you.
I'm a lot like you — unless I'm listening to actual music, yes, there's always some music or other, or a fraction of it, repeating in my head. Sometimes it's annoying and I have tricks to change the soundtrack. I think it's a variety of normal.
But I've also had a very intense version of it during periods of hypomania, where I can only describe it as mentally loud, insistent, and almost unbearable. I got to the point where I was crying and hitting my head against the pillow because "I'm walking on sunshine, whoa-oh, I'm walking on sunshine, whoa-oh, I'm walking on sunshine, whoa-oh, I'm walking on sunshine, whoa-oh, I'm walking on sunshine, whoa-oh, I'm walking on sunshine, whoa-oh, I'm walking on sunshine, whoa-oh, I'm walking on sunshine, whoa-oh" just would not come to a resolution or reduce in volume. At other times, I could have two or more pieces of music playing in my head simultaneously. But I was mad at the time, and even then I wouldn't describe my headmusic as psychosis.
As a separate thing, I've also had musical hallucinations (also not psychosis) and they're very different. You hear the music outside your head, with no indication that it's not happening in the outside world and being heard via your ears.