Let $M,N$ be two $n\times m$ matrices with $n\leq m$ and coefficients in an algebraically closed field of characteristic zero $K$, both of full rank $n$. 


Do there exist two upper triangular matrices $A\in SL(n)$ and $B\in SL(m)$ such that $A\cdot M \cdot B^{T} = \lambda N$ for $\lambda\in K\setminus\{0\}$ ?



Action of upper triangular matrices
I think the answer to your question is negative. Consider for instance
$$M = \begin{pmatrix}
1 & 0 \\ 
0 & 1
\end{pmatrix}, \quad N = \begin{pmatrix}
0 & 1 \\ 
1 & 0
\end{pmatrix}$$
Assume that there exist 
$$A = \begin{pmatrix}
a\_{11} & a\_{12} \\ 
0 & a\_{22}
\end{pmatrix}, \quad B = \begin{pmatrix}
b\_{11} & b\_{12} \\ 
0 & b\_{22}
\end{pmatrix}$$
with $\det(A) = \det(B) = 1$ and such that
$$A\cdot M\cdot B^{T} = N$$
Then
$$A\cdot B^{T} = \begin{pmatrix}
a\_{11}b\_{11}+a\_{12}b\_{12} & a\_{12}b\_{22}\\ 
a\_{22}b\_{12} & a\_{22}b\_{22}
\end{pmatrix} = \begin{pmatrix}
0 & 1\\ 
1 & 0
\end{pmatrix}$$
and hence either $a\_{22} = 0$ or $b\_{22}=0$ which contradict $\det(A) = \det(B) = 1$. 


More generally, your action stabilizes the locus of matrices of the form $\begin{pmatrix}
m\_{11} & m\_{12}\\ 
m\_{21} & 0
\end{pmatrix}$.


