We present systematic measurements of azimuthal anisotropy for strange and multi-strange hadrons (K0s, Λ, Ξ, and Ω) and ϕ-mesons at mid-rapidity (|y|< 1.0) in collisions of U+U nuclei at √sNN=193 GeV, recorded by the STAR detector at the Relativistic Heavy Ion Collider (RHIC). Transverse momentum (pT) dependence of flow coefficients (v2, v3, and v4) is presented for minimum bias collisions and three different centrality intervals. The results are obtained using the η sub-event plane method. These measurements are compared with the published results from Au+Au collisions at √sNN=200 GeV. Number of Constituent Quark (NCQ) scaling of the measured flow coefficients in U+U collisions is discussed. We also present the ratio of vn scaled by the participant eccentricity (εn{2}) to explore system size dependence and collectivity in U+U collisions. The magnitude of v2/ε2 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 (v3/v3/22, v4/v4/22) are studied as they are sensitive to the properties of the medium and mechanism of hadronization. The measured flow coefficients and the ratios between different flow harmonics are compared with hydrodynamic and transport model calculations.