Now you made me unsure, so skimmed https://www.youtube.com/watch?v=SrsWUmiayLM for a bit, which seems to me to confirm it's just horizontal camera movement. But there seems to be a zoom in/out functionality, which would move the camera in/out+vertically, but that's different than rotating around the character freely. The camera also does a ton of vertical movements by itself too, as Mario jumps/falls.
Yeah, I think you are actually right. I must have misremembered it. This source also suggests Mario Sunshine didn't have a "free" rotating camera but a combination of horizontal rotation and zoom: https://strategywiki.org/wiki/Super_Mario_Sunshine/Controls
Of course the question is whether there might have been an earlier game which had it. Regarding Ico, apparently it also allows only horizontal rotation (camera "panning" is horizontal movement): https://strategywiki.org/wiki/ICO/Controls
