I am curious. Let's say we are given two $n \times n$ matrices $G$ and $F$ where both $G$ and $F$ are nilpotent matrices of order $k$, i.e., $G^{k-1} \neq O$ and $F^{k-1} \neq O$, but $G^{k} =O$ and $F^{k} = O$ (where $O = [o_{ij}]$. Would $G \sim F$?
I don't think so. I have the examples
\begin{bmatrix}0 & 0& 1\\0 & 0& 1\\0 & 0& 0\\ \end{bmatrix}\begin{bmatrix} 5 & -3& 2\\ 15 & -9& 6\\ 10 & -6& 4\\ \end{bmatrix}
Both matrices square to zero. I couldn't find a invertible transformation $T$ such that $T^{-1}GT = F$, so I want to safely assume they are not similar.Thus I want to conclude that nilpotent matrices of the same other aren't necessarily similar.
Sorry for the rather short post. Thank you for your feedback.
In your example (and in fact any $3 \times 3$ example with order $2$), both have Jordan canonical form
$$ \pmatrix{0 & 1 & 0\cr 0 & 0 & 0\cr 0 & 0 & 0\cr}$$ so they are in fact similar. But in general, the answer is no. Consider $$ \pmatrix{0 & 1 & 0 & 0\cr 0 & 0 & 0 & 0\cr 0 & 0 & 0 & 1\cr 0 & 0 & 0 & 0\cr}\ \text{and}\ \pmatrix{0 & 1 & 0 & 0\cr 0 & 0 & 0 & 0\cr 0 & 0 & 0 & 0\cr 0 & 0 & 0 & 0\cr}$$