Let $\mathcal{S}_{\rho}$ be the subspace of eigenvectors corresponding to the eigenvalues $\lambda$ of $A$ such that $\rho(A)=|\lambda|$ and let $\mathcal{V}_1$ be the subspace of left singular vectors of $A$ associated with the maximal singular value $\sigma_1=\|A\|_2$ of $A$ (or, if you like, the eigenvector subspace corresponding to the eigenvalue $\sigma_1^2$ of $A^*A$).
We have that $\rho(A)=\|A\|_2$ if and only if $\mathcal{S}_{\rho}\cap\mathcal{V}_1\neq\{0\}$ (that is, the intersection of the two subspaces is not trivial).
Note that $\|Ax\|_2=\rho(A)\|x\|_2$ for all $x\in\mathcal{S}_{\rho}$ since $Ax=\lambda x$ for some $\lambda$ such that $|\lambda|=\rho(A)$ and $\|Ax\|_2=\sigma_1\|x\|_2$ for all $x\in\mathcal{V}_1$.
Also, $\|Ax\|_2<\sigma_1\|x\|_2$ for all $x\not\in\mathcal{V}_1$.
Hence if $\mathcal{S}_{\rho}\cap\mathcal{V}_1$ is nontrivial, then $\rho(A)\|x\|_2=\sigma_1\|x\|_2$ for some nonzero $x\in\mathcal{S}_{\rho}\cap\mathcal{V}_1$ and hence $\rho(A)=\sigma_1$.
On the other hand, if $\mathcal{S}_{\rho}\cap\mathcal{V}_1$ is trivial then any nonzero $x\in\mathcal{S}_{\rho}$ is not contained in $\mathcal{V}_1$ and thus $\rho(A)\|x\|_2=\|Ax\|_2<\sigma_1\|x\|_2$ for all nonzero $x\in\mathcal{S}_{\rho}$ resulting in $\rho(A)<\sigma_1$.
Let
$$
A=U\Sigma V^*
$$
be the SVD of $A$ and consider the partitioning
$$
A=[U_1,\tilde{U}]\begin{bmatrix}\sigma_1 I_{n_1} & 0 \\ 0 & \tilde{\Sigma}\end{bmatrix}\begin{bmatrix}V_1^*\\\tilde{V}^*\end{bmatrix},
$$
that is,
$$
AV_1=\sigma_1 U_1, \quad A\tilde{V}=\tilde{U}\tilde{\Sigma},
$$
where the columns of $V_1$ span the subspace $\mathcal{V}_1$. If $\mathcal{S}_{\rho}\cap\mathcal{V}_1$ is nontrivial, we can find a unitary transformation $M\in\mathbb{C}^{n_1\times n_1}$ such that $V_1M=[V_{\rho},V_{\rho}^{\perp}]$ where the columns of $V_{\rho}$ form an orthonormal basis of $\mathcal{S}_{\rho}\cap\mathcal{V}_1$ and $V_{\rho}^{\perp}$ spans the remainder of $\mathcal{V}_1$. Therefore
$$
AV_1M=A[V_{\rho},V_{\rho}^{\perp}]=\sigma_1 U_1 M = \sigma_1[U_{\rho},U_{\rho}^{\perp}],
$$
where the partitioning of $U_1M$ is conforming to the partitioning of $V_1M$. Since $Ax=\lambda x$ with $|\lambda|=\rho(A)$ for all $x$ in the range of $V_{\rho}$, we have that the columns $U_{\rho}$ and $V_{\rho}$ are related by $U_{\rho}=V_{\rho}D$, where $D=\mathrm{diag}(e^{i\phi_1},\ldots,e^{i\phi_k})$ (where $k$ is the dimension of $\mathcal{S}_{\rho}\cap\mathcal{V}_1$).
Therefore, in terms of SVD:
$\rho(A)=\|A\|_2$ iff $A$ has an SVD $A=U\Sigma V^*$ with $\Sigma=\mathrm{diag}(\sigma_1,\ldots,\sigma_n)$, $\sigma_1\geq\cdots\geq\sigma_n$, $U=[u_1,\ldots,u_n]$, $V=[v_1,\ldots,v_n]$, such that $u_j=e^{i\phi_j}v_j$, $j=1,\ldots,k$, for some $k>0$.
