Checking if two matrices are similar I have two matrices 
$$ \begin{pmatrix}
  2 & 1 & 0 \\
  0 & 2 & 0 \\
  0 & 0 & 3
 \end{pmatrix} $$
and $$ \begin{pmatrix}
  2 & 2 & 0 \\
  0 & 2 & 0 \\
  0 & 0 & 3
 \end{pmatrix} $$
They are not diagonalizable. Share the same characteristic polynomial, the same trace, same determinant, eigenvalues, rank. What could I use more to say if they are similar or not?

The first of your matrices is in Jordan canonical form, which must also be the Jordan canonical form of the second one (because the eigenvalue $2$ has algebraic multiplicity $2$ but geometric multiplicity $1$). So they are similar.
More concretely, we have
$$ \begin{pmatrix}
  2 & 1 & 0 \\
  0 & 2 & 0 \\
  0 & 0 & 3
 \end{pmatrix} = 
  \begin{pmatrix}1 \\ &2 \\ &&1 \end{pmatrix} 
\begin{pmatrix}
  2 & 2 & 0 \\
  0 & 2 & 0 \\
  0 & 0 & 3
 \end{pmatrix} 
  \begin{pmatrix}1 \\ &2 \\ &&1 \end{pmatrix}^{-1} 
$$
