Were you planning on using this asset in multiple forms (like on different backgrounds)? You could just promote the shape to a new layer and be done with it that way.
ON EDIT: Or make a selection, invert it and delete the background contents. More than one way anyhoo.
Image documents are rectangle Crops are rectangles. Some Image file formats support transparency so the image in these files these rectangle files may well not be a rectangle in shape.
The same thing is also true about layer the have rectangle bounds a width and a height. The background layer is and exception it doe not support transparencr. A background layer is always canvas size and has no transparency. Other layer can be any size larger or smaller then canvas
Save it as PNG and use the Trim command from Photoshop's Image menu.