TL;DR: yes, the video mode is cropped by approx. 1.28× (calculated by measurement). The effective video focal length is 36mm (in 35mm equivalent).
I read (in a user comment) that the iPhone's focal length is longer when shooting video because the frame is cropped slightly to enable video stabilization. Is this true?
It appears to be true that the video mode is cropped. I set up my iPhone 7 on a tripod aimed directly at a measuring tape, and took both a photo and and video of the tape. I did not move my phone or the measuring tape between shots.
Width measurement of iPhone 7 photo mode
Width measurement of iPhone 7 video mode (exported still frame)
From these images, I estimate the width of the photo mode wP = 49.6 cm, and the width of the video mode wV = 42.2 cm.
And if so, how can we calculate the "new" focal length that takes this cropping into account?
If we think of the video mode as its own crop factor with respect to the photo mode, then we only need to compute how much the image circle was reduced by as a result of cropping. The image circle of an image is just the cicumcircle, or smallest circle that will fit around a shape.
If the video image were cropped by the same factor in both dimensions (i.e., keeping the same aspect ratio), then we could just divide the measured image widths. But because the aspect ratio also change, we must compute the diameters of the images.
Video mode sized and overlaid on photo mode, with image circles and measurements
Recall that Pythagoras told us that the diameter of a rectangle is given by d = √(w² + h²). But I didn't measure height. However, we do know height relative to width, from the aspect ratios: h = w/A.
Putting them together, we have d = (w/A) * √(A² + 1).
The aspect ratio of the photo mode is 4032:3024 = 4/3. The aspect ratio of the video mode is 1920:1080 = 16/9.
Dividing the diagonal of the photo mode (dP) by the diagonal of the video mode (dV), I calculate the video mode's image circle is 1.28 times smaller than the photo mode's image circle.
This diagonal crop ratio is exactly what crop factor is, and is applied to focal lengths (as a multiplier) to get "35mm equivalent focal length".
So, since the iPhone 7's photo-mode uses the lens's full area, and that lens provides a field-of-view equivalent the FoV of a 28mm lens on a 35mm format body, then the iPhone 7's video-mode FoV is equivalent to 1.28 * 28 ≈ 36mm lens (in 35mm equivalent).