Are two representations $V, W$ isomorphic if $\left<\chi_V, \chi_W\right> > 1$ for the inner product of their characters?

Question

I know from Schur's lemma(s) that if $V, W$ are two isomorphic irreducible representations of a group $G$, then $\left<\chi_V, \chi_W\right> = 1$. This is because the dimension of the space of intertwining maps between $V$ and $W$ is equal to $\left<\chi_V, \chi_W\right>$. Then, what if $\left<\chi_V, \chi_W\right> > 1$? Then we know that there are multidimensional intertwining maps between $V$ and $W$. Does it then follows that $V \cong W$?