Algorithm for identifying polygons (set of points)
I'll assume you are given two sets of points and want to determine whether one can be transformed to match the other.   This is the pointset registration problem in computer vision.  There are standard algorithms for this problem.

One of the crucial challenges is outliers -- e.g., points in the first set that don't have any corresponding point in the second set, or that have a large amount of error in their location.  If you don't have outliers in your setting, then the problem becomes noticeably easier: simple least-squares linear regression might suffice.

If you have to deal with outliers, one standard way is to use RANSAC to find a rigid transformation that, when applied to the first set, makes it match the second set.  A rigid transformation is determined by the correspondence between two points.  So, you can use RANSAC: repeatedly pick a pair of points in the first set, choose a candidate pair of points in the second set for them to be matched to, compute the rigid transformation that maps the first two points to the latter two points, apply it to all points in the first set, find all inliers (points that are well-matched), update your estimate of the rigid transformation, count the number of inliers, and do this many times and keep the best rigid transformation you've found.

There are many other approaches as well.  If you do some research, you should be able to find many algorithms, techniques, and even existing tools.