Suppose you have 4 matrices with singular value decompositions
$A = U_1 \Sigma_A V_1^{\dagger}$, $B = U_2 \Sigma_B V_2^{\dagger}$, $C = U_1 \Sigma_C V_1^{\dagger}$ and $D = U_2 \Sigma_D V_2^{\dagger}$.

Are the singular value decompositions of $A+B$ and $C+D$ of the form
$A+B = U_3 \Sigma_{A+B} V_3^{\dagger}$ and $C+D = U_3 \Sigma_{C+D} V_3^{\dagger}$?