1. A wear amount measuring device comprising: a computer including a storage device having a wear amount measuring program stored therein, and a processor configured to execute the wear amount measuring program; and an image display device operatively coupled to the computer and configured to display an image based on a display signal received from the computer, the wear amount measuring program being configured to cause the image display device to display a real object image of a replaceable component based on real object image data containing a real wear amount measurement target and a real reference part of the replaceable component, the real reference part functioning as a measurement-related positioning reference for the real wear amount measurement target, and display a plan image based on design plan data corresponding to the replaceable component in an unused state, the design plan data containing a plan wear amount measurement target and a plan reference part, the plan wear amount measurement target and the plan reference part corresponding to the real wear amount measurement target and the real reference part, respectively, the wear amount measuring program including an image processing section configured to execute an image processing of overlapping the real object image and the plan image at an equal scale on a corresponding positional relation when the real reference part in the real object image and the plan reference part in the plan image are matched, and a wear amount computing section configured to compute a wear amount based on a magnitude of an interval between a measurement contour line drawn along an outer peripheral contour of the real wear amount measurement target in the real object image and a plan outer peripheral contour line in the plan image.