We present measurements of the linear polarization amplitudes and the strong relative phases that describe the flavor-untagged decays $B_{d}^{0}\to J/\psi K^{*0}$ and $B_{s}^{0}\to J/\psi\phi$ in the transversity basis. We also measure the mean lifetime $\bar{\tau}_{s}$ of the $B_{s}^{0}$ mass eigenstates and the lifetime ratio $\bar{\tau}_{s}/\tau_{d}$. The analyses are based on approximately 2.8 fb$^{-1}$ of data recorded with the D0 detector. From our measurements of the angular parameters we conclude that there is no evidence for a deviation from flavor SU(3) symmetry for these decays and that the factorization assumption is not valid for the $B_{d}^{0}\to J/\psi K^{*0}$ decay.