Define \begin{align*} A&=\begin{bmatrix}1&1\\0&1\end{bmatrix} & B&=\begin{bmatrix}1&0\\0&1\end{bmatrix} \end{align*} and check that \begin{equation*} \charpoly{A}{x}=\charpoly{B}{x}=1-2x+x^2=(x-1)^2 \end{equation*} and so $A$ and $B$ have equal characteristic polynomials. If the converse of Theorem SMEE were true, then $A$ and $B$ would be similar. Suppose this is the case. More precisely, suppose there is a nonsingular matrix $S$ so that $A=\similar{B}{S}$.
Then \begin{equation*} A=\similar{B}{S}=\similar{I_2}{S}=\inverse{S}S=I_2 \end{equation*}
Clearly $A\neq I_2$ and this contradiction tells us that the converse of Theorem SMEE is false.