We explore synchronization transitions in a generalized Kuramoto model with $n$-dimensional vector oscillators on complete graphs (CG) and on $d$-dimensional lattices. On the CG, we analytically derive universal critical exponents $(\beta ,\bar{\nu })=(1/2,5/2)$ for all even $n$. Through the Botet–Jullien–Pfeuty relation, the CG exponent $\bar{\nu }=5/2$ predicts an upper critical dimension $d_u=5$ for the corresponding lattice models. For locally coupled systems with $d<d_u$, we identify frequency entrainment for arbitrary $n$ using a newly developed entrainment connectivity matrix, and characterize the transitions with two complementary order parameters. Using spin-wave theory together with numerical simulations, we obtain critical exponents across the $(n,d)$ parameter space. These results suggest a family of $n$-independent but $d$-dependent universality classes, offering a unified perspective on how symmetry and dimensionality shape synchronization phenomena.