Are They Similar Polygons?
Mathematica, 97 bytes (SBCS)



Or@@(Equal@@(a@u/#)&amp;/@(a/@NestList[RotateLeft,v,Length@v]))
a@l_:=Norm[#〚1〛-#〚2〛]&amp;/@l~Subsets~{2}


You can try it online!

The function a computes all the pairwise distances between any two vertices of the polygon. 

The main function applies this to the first list and to every rotation of the second list, to try and find the corresponding vertices. The rotation of the second list is correct if all the corresponding distances have been scaled accordingly.

Thanks to @J42161217 for saving me some 3 bytes