EDIT: The original answer, included above the image, only considers cases E and F. A more complete answer follows the image.
The question can be simplified a little further from my earlier comment: for which $k$, $m$ and $n$ can a rectangle made up of $k$ squares fit into an $m \times n$ rectangle in only way? (Here "way" means the shape of the small rectangle and relative orientation of the small and big rectangles.)
Write factorisations of $k$ as $k=rs$, where $r \leq s$, and assume $m \leq n$. The condition is then that there is a unique factorisation of $k$ with $r \leq m$ and $s \leq n$ (so that the small rectangle fits inside the big rectangle), and if $m < n$ then $s > m$ (so that it can't be rotated if the big rectangle is not a square).
The image shows a not quite complete (it does not include, for instance, the case $k=mn$) list of the ways in which two identical rectangles can overlap. I'll give a couple of examples to show how it goes.
Cases A and B are the possibilities for squares. For $n\times n$ squares, the intersection can take size $k$ if and only if $k$ has a unique factorisation with $r, s \leq n$.
Now assume $m < n$. If we have $k=mx$ for some $x \leq m$ then C and either G or H provide distinct configurations with overlap $k$.
The rest of the restrictions on $k$, $m$ and $n$ can be worked out by a rather tedious case check: I don't expect there's a particularly nice phrasing of the conditions you'll obtain.