The two-dimensional triangular-lattice antiferromagnet (TLAF) is a textbook example of frustrated magnetic systems. Despite its simplicity, the TLAF model exhibits a highly rich and complex magnetic phase diagram, featuring numerous distinct ground states that can be stabilized through frustrated next-nearest-neighbor couplings or anisotropy. In this paper, we report low-temperature magnetic properties of the TLAF material CsCeSe₂. The inelastic neutron scattering (INS) together with specific heat measurements and density functional theory calculations of crystalline electric field suggest that the ground state of Ce ions is a Kramers doublet with strong easy-plane anisotropy. Elastic neutron scattering measurements demonstrate the presence of stripe-yz magnetic order that develops below T_N=0.35K, with the zero-field ordered moment of m_Ce≈0.65μ_B. Application of magnetic field first increases the ordering temperature by about 20% at the intermediate field region and eventually suppresses the stripe order in favor of the field-polarized ferromagnetic state via a continuous quantum phase transition (QPT). The field-induced response demonstrates sizable anisotropy for different in-plane directions, 𝐁∥𝐚 and 𝐁⁢⊥⁢𝐚, which indicates the presence of bond-dependent coupling in the spin Hamiltonian. We further show theoretically that the presence of anisotropic bond-dependent interactions can change the universality class of QPT for 𝐁∥𝐚 and 𝐁⁢⊥⁢𝐚.