We present systematic measurements of azimuthal anisotropy for strange and multistrange hadrons (K-s(0), Lambda, Xi and Omega) and phi mesons at midrapidity (vertical bar y vertical bar < 1.0) in collisions of U U nuclei at root S-NN = 193 GeV, recorded by the STAR detector at the Relativistic Heavy Ion Collider. Transverse momentum (p(T)) dependence of flow coefficients (v(2), v(3), and v(4)) is presented for minimum bias collisions and three different centrality intervals. Number of constituent quark scaling of the measured flow coefficients in U + U collisions is discussed. We also present the ratio of v(n) scaled by the participant eccentricity (epsilon(n){2}) to explore system size dependence and collectivity in U + U collisions. The magnitude of nu(2)/epsilon(2) is found to be smaller in U + U collisions than that in central Au + Au collisions contradicting naive eccentricity scaling. Furthermore, the ratios between various flow harmonics (v(3)/v(2)(3/2), v(4)/v(2)(4)(/2)) are studied and compared with hydrodynamic and transport model calculations.