Solution:

Let $O$ be an arbitrary point in the plane and let $A_{1,1} A_{2,1} \ldots A_{n,1}$ be an $n$-gon similar to $\mathcal{P}$ and containing $O$. Consider the $n$-gons
$$O A_{1,1} A_{1,2} \ldots A_{1, n-1},\quad O A_{2,1} A_{2,2} \ldots A_{2, n-1},\quad \ldots,\quad O A_{n,1} A_{n,2} \ldots A_{n, n-1}$$
that are similar to $A_{1,1} A_{2,1} \ldots A_{n,1}$ and have the same orientation. Using a rotation and a homothety with center $O$ we see that the $n$-gon $A_{1, j} A_{2, j} \ldots A_{n, j}$, $j=2, \ldots, n-1$, is similar to $A_{1,1} A_{2,1} \ldots A_{n,1}$ and has the same orientation.

Denote by $o$ and $a_{i, j}$ the numbers assigned to the points $O$ and $A_{i, j}$, respectively. Summing up the equalities
$$o+\sum_{j=1}^{n-1} a_{i, j}=0,\quad i=1, \ldots, n$$
and using that
$$\sum_{i=1}^{n} a_{i, j}=0,\quad j=1, \ldots, n-1$$
we get $n o=0$, which implies the assertion.