The first (v_{1}^{fluc}), second (v_{2}), and third (v_{3}) harmonic coefficients of the azimuthal particle distribution at midrapidity are extracted for charged hadrons and studied as a function of transverse momentum (p_{T}) and mean charged particle multiplicity density ⟨N_{ch}⟩ in U+U (sqrt[s_{NN}]=193 GeV), Au+Au, Cu+Au, Cu+Cu, d+Au, and p+Au collisions at sqrt[s_{NN}]=200 GeV with the STAR detector. For the same ⟨N_{ch}⟩, the v_{1}^{fluc} and v_{3} coefficients are observed to be independent of the collision system, while v_{2} exhibits such a scaling only when normalized by the initial-state eccentricity (ϵ_{2}). The data also show that ln(v_{2}/ϵ_{2}) scales linearly with ⟨N_{ch}⟩^{-1/3}. These measurements provide insight into initial-geometry fluctuations and the role of viscous hydrodynamic attenuation on v_{n} from small to large collision systems.