The development of nanomaterials with multienzyme activity for advanced sensing and biosensing assays has attracted attention. In this study, a Cu−Co bimetallic nitrogen-doped carbon catalyst (CoCu@NC) was synthesized. The prepared nanomaterials exhibit catalase- and oxidase-like mimicking activities by adjusting the pH. The catalase-like activity of the CoCu@NC was investigated by quenching of terephthalic acid (TA) fluorescence at pH 11 in the presence of H2O2, while its oxidase behavior was confirmed by oxidation of 3,3′,5,5′-tetramethylbenzidine (TMB) as chromogenic substrate in the presence of O2 at pH 3. Furthermore, CoCu@NC’s oxidase-like activity was used successfully to detect hydroquinone (HQ) at a concentration range of 1−900 nM with a detection limit of 0.22 nM and the anticancer drug doxorubicin (DOX) with a wide linear response ranging from 5 fM to 200 pM and an exceptionally low detection limit of 1.66 fM by reduction of oxTMB to TMB. DOX interacts in situ with single-stranded (ssDNA) and double-stranded DNA (dsDNA), reducing the quinone ring in its structure to hydroquinone (HQ) and oxidizing guanine bases to 8-oxoguanine. Based on this phenomenon, we designed a label-free colorimetric sensor for measuring DNA damage (ranging from 1 pM to 1 μM), in which this sensor operates by the disappearance of the blue oxTMB solution and the presence of the DNA/DOX. Furthermore, this designed sensor is sensitive to the number of guanine bases in ssDNA and dsDNA. As the number of guanine bases (1−12) in DNA sequences increases, a greater color change is observed. Finally, in the presence of H2O2-induced DNA damage, no intercalation occurred between DOX and the DNA-damaged sequences, with the color change observable with the naked eye. Therefore, this visualization assay demonstrates a low-cost, simple, rapid, sensitive, and effective method for detecting DOX drug and damaged DNA. Additionally, CoCu@NC magnetic nanostructures could be easily recollected and reused by applying a magnetic field.