We performed a comprehensive electron spin resonance, magnetization, and heat capacity study on the field-induced magnetic phase transitions in the kagome antiferromagnet α-Cu3.26Mg0.74(OH)6Br2. With the successful preparation of single crystals, we mapped out the magnetic phase diagrams under the c axis and ab-plane directional magnetic fields B. For B∥c, three-dimensional (3D) magnon Bose-Einstein condensation (BEC) is evidenced by the power law scaling of the transition temperature, Tc∝(Bc-B)2/3. For B∥ab, the transition from the canted-antiferromagnetic state to the fully polarized state is a crossover rather than a phase transition, and the characteristic temperature has a significant deviation from 3D BEC scaling. The different behaviors of the field-induced magnetic transitions for B∥c and B∥ab could result from the Dzyaloshinkii-Moriya (DM) interaction with the DM vector along the c axis, which preserves the c-axis directional spin rotation symmetry and breaks the spin rotation symmetry when B∥ab. Our findings have the potential to shed light on the investigations of magnetic anisotropy on the field-induced magnon BEC in a quantum antiferromagnet.