May I ask what makes you expect that the size would be smaller?
With docker save the metadata of the image, the tag points to, and every image layer it consists of are saved. If you save a single image the only way to bring the size down is a) optimize your image to be smaller and b) use gzip (or even better pigz) to create a tar.gz instead of an uncompress tar archive.
You do benefit from image layer de-dupplication if you save multiple tags at once that share the same parent images. There is no cherry picking of image layers in vanilla docker.
Though there is a docker fork, designed for embedded system that indeed creates image that only contain the diff from one tag to another. Someone posted it in the forum, but I don’t recall the name (update: see balenaEngine - A container engine purpose-built for IoT devices)