We investigate tank-treading and breathing dynamics of individual star molecules under shear flow and their relation with the macromolecule architecture. Tank-treading consists of the rotation of the arms of the star around a molecule's center, whereas breathing consists in expansions and contractions of the whole molecule at a certain characteristic frequency. We derive scaling arguments for the trends of the frequency and decorrelation rates of both rotation and breathing modes versus the shear rate <(gamma)over dot>, which are supported by extensive Brownian hydrodynamics simulations. We find that breathing occurs if <(gamma)over dot> is made faster than the equilibrium decorrelation rate Gamma(0) of two perpendicular eigenvectors of the gyration tensor. Gamma(0) increases with the star functionality, F, as Gamma(0) approximate to F-0.5, which contrasts with the rotational and arm-length relaxation rates (F-0.6 and F-0.15 respectively). For <(gamma)over dot> > Gamma(0), the star becomes ellipsoidal and elongates in the flow direction and this determines the onset of the non-Newtonian regime. Remarkably, <(gamma)over dot>/Gamma(0) provides universal trends for the shape, dynamics, and rheology of the star polymer.