For the most part, yes.
"... there's no 'but'... They both share the same term because the both squash the image horizontally to store a wide-image in a regular-frame. Is film a lossless process, but DVD the information is lost?"
The problem is, you keep thinking that the "video" is squeezed horizontally, and it's not. You can't "squash" the pixels - you can only stretch them. The width of the video as it is stored on the DVD is the width of the video. If you were to "squeeze" the video to fit it into the space on the DVD, you would have to lose lines of video (or they would be combined with their neighbors.) The pixels themselves remain the same size regardless, so to answer your question: yes, anamorphic does not lose resolution when squeezed, because film has a tremendously higher resolution (i.e. smaller "pixels", if you will), than video, which is only 720x480.
What DVD does to get around this is increase the vertical resolution. Since the pixels can't change shape, you add more pixels to the other dimension. Therefore, for a 4x3 image, one of every four lines are ignored, so that the vertical image is in the correct aspect ratio of the horizontal image. However, when the image is displayed on a 16x9 set, all of the vertical lines are displayed, and the horizontal pixels are wider to make the correct aspect ratio.
Point being, nothing is squeezed into a frame in a DVD transfer. It's quite the opposite, since more vertical lines of visual information are added, which is why anamorphic transfers take up more space than letterbox transfers.
Film, on the other hand, takes the same image and does squeeze it into a smaller frame.
Film squeezes the same picture information horizontally, and DVD adds picture information vertically.Thus, while both appear to make people tall and skinny, it is for very different reasons.
Whew, I hope that helped.
