Photoshop save selection not working

I am not sure what you are trying to do. Are you trying to fill the thin white strip at the bottom with black like the rest of the background or are you trying to select the green bordered area and have that be your full image?

If you make a selection using any selection tool and then go under the Image menu and choose Crop, the image becomes your selected area and the rest is cropped off.

If you make a selection and then go under Select and choose Save Selection, you are just saving the selected area as a new Channel into the same image. And yes, in this case the white area would still be there because you have not really cropped anything or changed anything in the image.

If you want everything outside your selected area to be a different color, you can make your selection, then go under the Select menu and choose Select Inverse. This "selects" everything but what you had before. Then you can go under Edit and choose Fill and fill the "inverted selection" with another color - like black in this instance.

Since the sliver of white is quite narrow, enlarge the view (strike Cmd + several times) and then click in the area with the Magic Wand (Tolerance in the Options bar set to 8px, checkmark Contiguous) to select the white and strike the Delete key. Then return to your previous view (Cmd -). followed by Image > Trim > Transparent pixels.

After you make a selection then use Image>Crop. Your document will be the size of the selection.