It's a standard feature in nearly all common video editors (i.e. DaVinci Resolve or Adobe Premiere).
Usually, stabilization goes over all video frames and tries to find image transformations (rotation + translation + zoom) that make a frame match as closely as possible with the previous frame. That's an oversimplified explanation, but from a user point-of-view, these tools are mature enough to be applied with just a few clicks.
This video is definitely the result of that, as, whoever did it, didn't even bother to insert a cut when the feed switches between left side and right side camera, thus making the stabilization spazz out momentarily.
Where do you see the camera switch side? The fin on which the camera is mounted moves every now and then, but I think we only ever see the perspective of one camera.
There used to be a great subreddit for image stabilisations, always enjoyed seeing stuff from there and people would sometimes go into detail about the tools and techniques they used .
There probably still is, but there used to be too.
Most advanced video compositors/editors have stabilization features. What you do is you give the program a couple of reference points that you know aren't moving, and the program will automatically track those points over time