What is this style of photo looping called?
According to Know Your Meme:
Wigglegrams (also known as stereographs) are animated images that simulate 3D effect by looping two or three frames of an object shot from the vantage points of the viewer’s right and left eyes. Similar to cinemagraphs, each instance usually consists of a few continuous frames that play in loop.
With just two images (stereoscopy), it's called Wiggle Stereoscopy (stereo meaning two). For the general case (two or more), the general term is just Wiggle 3-D
From that wiki page:
Wiggle stereoscopy is an example of stereoscopy in which left and right images of a stereogram are animated. This technique is also called wiggle 3-D or wobble 3-D, sometimes also Piku-Piku (Japanese for "twitching").
Three or five images with good parallax may produce a better effect than simple left and right images.