If you consider an image as a grid of pixels with x,y coordinates, by georeferencing you are essentially telling the software that coordinate p in the image should really be at coordinate q in the GIS data CRS.
This concept can also be applied to images under the term registration. Two images can be said to be co-registered if the x,y pixel coordinates represent the same point in both images. That means that the same intersection between two lines (for example) occurs at the same pixel coordinate in both images. This is often an issue in color print presses as each color is applied separately, and when the different colors aren't co-registered you get blurry images.
When images are produced by scanning, it is very difficult to get them to co-register due to distortions, misaligned feeding, or any number of other issues. Images can be produced that are co-registered either by post processing or using identical output settings in a software and simply changing what is or isn't visible when printing the same extent (like hitting print with your map open but turning layers off and on in between while not changing any other settings).
As long as your images are co-registered, you could theoretically use the same link table for multiple images, since you'd be transforming between just two coordinate systems (all the image x,ys are the same). Otherwise, as is most likely the case, you cannot use the links from one image to georeference another since each image will have a different coordinate value for the same features. The common point in the images you are trying to reference might be at 3,4 in one and 8,7 in the other. If you say one of them should match 5,5 in the CRS you are georeferencing to, the other will of course be wrong with the same transformation.
Bottom line, you'll probably need to georeference each image independently for best results. Note as pointed out in the comments that you need to save the transformation on the image once done (or before switching to another image) for it to be loaded correctly in the future by using the Update Georeferencing command. This actually updates the image rather than just storing a list of transformation control points.