Well you could determine the minimum area bounding rectangle and determine the ratios of their diameters. Or you could do the aforementioned and determine a minimum area bounding circle and determine their ratios. It really depends upon what kind of metric you want...there are lots (ie perimeter, area diagonal base etc etc)
This extension doesn't require any license level http://www.arcgis.com/home/item.html?id=564e2949763943e3b9fb4240bab0ca2f
do you have an image of what you are trying assess similarity for? rectangular building would be easily assessed with minimum area bounding rectangles. You can also accompany that with area and/or perimeter ratios. Shape similarity can be described on a variety of levels
as for shape congruence specifically, you can determine interior angles of polygons as well as segment lengths, and hence ratios of those and area, but there are no built-in tools to do so.
You could convert polygons to raster (feature to raster) and run "zonal statistics".
Join the result zonal table with feature and sort by the "sum" column.
This give the result you want.
The smaller pixel value in the conversion, gives better results.
I think if you look for features with a similar width and height of a minimum bounding rectangle, area, and perimeter length - you'd be close. A polygon with those 4 similarities is likely to be the same shape. But I think you'd have to code or build the model yourself.