Suppose I have two $n$-sided polygons A and B. Is there a non-trivial upper bound on the number of parameters (eg. area, perimeter, etc) of the two polygons, that need to be the same, for A and B to be identical? 
(For example, if we consider a simpler case to the problem when say A is constrained to lie inside B, then if area(A) = area(B), then A and B are identical. Hence area is the only non-trivial parameter in this case)



Uniqueness of a polygon
I guess you are looking for a nice answer, but here is a stupid one.
On the other hand I'm sure that there is no "nice answer".


A polygon can is uniquely determined by length of sides $\ell\_i$ and angles $\alpha\_i$.
Thus we have to find a complete set of invariants for sequence $(\alpha\_1,\ell\_1,\alpha\_2,\ell\_2,\dots\alpha\_n,\ell\_n)$ which survive after even cyclic shifts and reversing order.


Then you prepare [symmetric polynomials](https://en.wikipedia.org/wiki/Ring_of_symmetric_functions) for your group.
Say take all monomials of degree at most one in each $\alpha\_i$ and $\ell\_i$ and take its mean value it along the group.
You obtain a big collection of polynomial expressions in $\alpha\_i$ and $\ell\_i$ which gives complete invariant (perimeter will be one of them).


