Focus is focus is focus, whether for still or video. But the story does not end there. Focal length, aperture, sensor size, circle of confusion, etc., all play a role in determining whether you perceive something to be in focus or not. Here's a link to the math.
The long and short of it is that quite often a camera that shoots both stills and video will use a larger sensor area for stills and a smaller subset of the sensor for video. Those math equations will tell you that smaller sensors have more things in focus than larger sensors. A 35mm full-frame camera has a 42mm image circle, which is huge next to the 6mm image circle of a typical prosumer camera, which is in turn larger than the image circle of the typical smartphone.
On top of that, yes, smartphones also dynamically focus the image, both for still and video images. You probably don't see the refocusing happening in video because the depth of field is already so deep, but it happens.
There are video cameras that have sensors as large as full-frame still cameras; they are just as difficult to focus as a still camera, and have just as shallow a depth of field for the same image size, focal length, aperture, and circle of confusion.