This doesn't require multiple cameras at all. It requires multiple (>1) photos of the same subject/setting from slightly different angles. If you have a subject that is moving fast you need a higher shutter speed and a quick succession of photos in order to build this but it's doable. But as you allude to ideally you could take the photos precisely at the same time using multiple cameras. It's really not that much unlike how The Matrix effect was done really.

Doesn't necessarily need a smartphone, but you'd need multiple cameras taking the picture at once if anything in the pictures is moving.

I think it'd be better if the cameras are parallel to each other.

I'm not sure how the reconstruction from the depth map works, though. That looks cool.