Specific heat and ac magnetic susceptibility measurements, spanning low temperatures ($T ≥ 40$ mK) and high magnetic fields ($B ≤ 14$ T), have been performed on a two-dimensional (2D) antiferromagnet Cu(tn)Cl$_{2}$ (tn = C$_{3}$H$_{10}$N$_{2}$). The compound represents an $S = 1/2$ spatially anisotropic triangular magnet realized by a square lattice with nearest-neighbor ($J/k_{B} = 3$ K), frustrating next-nearest-neighbor ($0 < J^{\prime}/J < 0.6$), and interlayer ($|J^{\prime \prime}/J| ≈ 10^{-3}$) interactions. The absence of long-range magnetic order down to $T = $ 60 mK in $B = 0$ and the $T^{2}$ behavior of the specific heat for $T ≤ 0.4$ K and $B ≥ 0$ are considered evidence of high degree of 2D magnetic order. In fields lower than the saturation field, $B_{\text{sat}} = 6.6$ T, a specific heat anomaly, appearing near 0.8 K, is ascribed to bound vortex-antivortex pairs stabilized by the applied magnetic field. The resulting magnetic phase diagram is remarkably consistent with the one predicted for the ideal square lattice, except that $B_{\text{sat}}$ is shifted to values lower than expected. Potential explanations for this observation, as well as the possibility of a Berezinski-Kosterlitz-Thouless (BKT) phase transition in a spatially anisotropic triangular magnet with the Néel ground state, are discussed. Comment: 8 pages, 9 figures