In the aligned two-Higgs-doublet model, the alignment of Yukawa matrices in flavour space guarantees the absence of tree-level flavour-changing neutral currents, while allowing at the same time for new sources of CP violation, implying potentially large effects in many low-energy processes. In this work we study the constraints from exclusive radiative $B\to V\gamma$ decays, where $V$ denotes a light vector meson. The current experimental data on the CP-averaged branching ratios and the direct CP and isospin asymmetries are analyzed. It is found that, while the branching ratios and direct CP asymmetries do not constrain the parameter space much further compared to the inclusive $B\to X_{s,d} \gamma$ decays, complementary constraints can be obtained from the isospin asymmetries $\Delta(K^*\gamma)$ and $\Delta(\rho\gamma)$. In addition, correlations between the various observables in exclusive $B\to V\gamma$ and inclusive $B\to X_{s,d} \gamma$ decays are investigated in detail, and predictions are made for several so far unmeasured observables.