I just looked at a few videos and it seems that the analog stick does indeed move the camera, though apparently only horizontally, not "freely" with arbitrary rotations. I'm not sure though.

How is that different from Super Mario Sunshine? It was probably two decades ago I last played it, but I think it was the same.

No there you can do arbitrary camera rotations, not just horizontal ones. For example, you can view the character from above.

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

I guess then the first game I definitely know that had a free camera was The Legend of Zelda: The Wind Waker, which came out a few months after Sunshine. This is also confirmed here: https://strategywiki.org/wiki/The_Legend_of_Zelda:_The_Wind_...

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