Nanozymes based on metal-organic frameworks (MOFs) have been concentrated on due to their naturally high-disperse metal active sites and the adjustable coordination chemistry. In this work, an N-rich melamine (Mel) was introduced into the Cu-MOF composed of copper(II) nitrate and 2-aminoterephthalic acid (Cu-NH₂-BDC-Mel) to mimic the laccase, which enzyme-like activities were assessed and applied in sensing analyses toward several phenols and amines. Compared to unmodified Cu-NH₂-BDC, the resulting Cu-NH₂-BDC-Mel exhibits enhanced laccase-like activity, superior stability and catalytic kinetics. It is demonstrated that melamine-doping has increased nitrogen content as well as the surface area, as a result, exhibits a lower Michaelis–Menten constant (K_m) (0.1877 mM) and higher maximum reaction rate (V_max) (1.7933 × 10⁻³ mM·min⁻¹) in comparison with that of natural laccase. Based on that, an efficient colorimetric sensing strategy for several phenols and amines was built up with excellent selectivity and anti-interference by using the laccase-like Cu-NH₂-BDC-Mel, the detection limits are 3.51 μM of adrenaline and 4.41 μM of dopamine. The work broadens the prospect development of bio-colorimetric sensing based on the ligand-modified Cu-MOFs nanozymes catalysis.