Answer
False
Work Step by Step
Consider two matrices A and B have the same set of eigenvalues
$A=\begin{bmatrix}
1 & 0\\
0 & 1
\end{bmatrix}\\
\rightarrow \lambda_1=\lambda_2=1$
and $A=\begin{bmatrix}
1 & 1\\
0 & 1
\end{bmatrix}\\
\rightarrow \lambda_1=\lambda_2=1$
We can notice that both matrices have eigenvalue $\lambda =1$ with multiplicity 2 but they are not the same, $A$ is not similar to $B$.
Hence, the statement is not true.
