No dog was ever on the ISS.
This is from an advert of a Japanese cell phone provider called SoftBank. Here's the full video:
The ad is part of a series. This episode is called "父と交信" or (Communicating with Dad; 父/dad being the dogs name). I found no hint that the production was in any way extraordinary.
It appears to be filmed on a green screen background. My guess is that the body is a puppet and the head was later composed in. For the paddling shot, it may be that the space suit hides a harness that the dog was kept in. From walking a dog in a harness once I remember that if you lift it up by the harness and keep it close to the ground it may do the dog paddling shown.