I have in mind not simply plane figures, but surfaces in 3-space. A case in point is the surfaces of the headlights of a vehicle, and the surfaces of its tail lights. There is obviously a correlation between them, although they are not similar in the strict geometric sense of the word. I would be surprised if some definite analytics/metrics hadn’t been applied, at least partially, to this question, and so I am tagging this as a reference-request.
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There are many possible ways to compare shapes. Procrustes analysis is one of the simplest; a point distribution model is another simple one. Both use a chosen set of features to determine similarity.
A "correlation coefficient," properly speaking, is a measurement of the linear relation between two variables -- it's a very specific measure of similarity. Since there doesn't seem to be any indication that we expect the relation between various points on the surface of a headlight to be linear, I'm going to guess that you're actually asking about similarity measures in general.
The answer in this case is that there are lots of similarity measures between shapes, and different ones are better for different situations. Some of the main applications are in video compression, where we want to compress 2D images in a human-friendly way) and medical imaging (where we want to find topological features of 3D objects).
I'm not an expert by any means, but Googling "3D similarity measures" brings up several papers describing different similarity measures for 3D shapes in various contexts. You could start by looking through those and seeing if anything looks like what you want.