Yes, only a portion of the sensor is used for recording video (or photos) at a different aspect ratio than that of the sensor. This is done so that you can use the same lenses for video as for photos. By video standards, 1080p should be 1920 by 1080 with square pixels. The physical pixels on the sensor are square, so we can't change the pixel aspect ratio (which refers to the actual shape of the pixels.
This means our only options to get a 1920 by 1080 image are either to crop our sensor or use an anamorphic lens. Anamorphic lenses are a class of lens that do not resolve an image equally in both direction, but instead squeeze a wider width in to a smaller space. This was originally used for shooting anamorphic wide screen on traditional film, but there is no reason the same technique wouldn't work on a DSLR if you get it to make use of the extra sensor space.
There isn't a whole lot of advantage to doing this though. You might be able to catch a little tiny bit more light for low light performance, but you'd probably lose more quality in the anamorphic optics and having to use custom firmware to handle shooting anamorphic video. You couldn't simply capture all the data and post process as the data rate that the camera can maintain for video is not high enough to record non-stop full resolution images from the sensor, so it would have to be processed to a 1920 by 1080 image (or nearly that) in camera.
Since the sensor is still much higher resolution than the finished video resolution, the cropping is really not a significant impact on final quality.