You can not save it as a video, but you can save the frames to a folder.
To do this, you need to use the output_file.so plugin.
You can run mjpg-streamer -o "output_file.so --help" to get some info about what parameters you can use (probably -m and -d)
If you want to stream and save the file at the same time, you would need to edit the service file of mjpg-streamer and start it with somethink like:
There is no settings page. You can use the input_uvc.so plugin to set stuff like resolution, FPS and JPEG quality.
run this to get the available options:
mjpg_streamer -i "input_uvc.so --help"
As alternatives you can use ffmpeg (also no settings page, you do everything from CLI like with mjpeg-streamer) or if you want a GUI, I would stick to motion.