We have studied the nuclear region of the previously detected dual AGN system in the galaxy pair IRAS 05589 + 2828 and 2MASX J06021107 + 2828382, through new optical spectroscopy observations, along with radio and X-ray archival data. Our multiwavelength data strongly suggest that the Sy1 IRAS 05589+2828 (z = 0.0330 ± 0.0002) conforms to a dual AGN system with the Sy2 2MASX J06021107 + 2828382 (z = 0.0334 ± 0.0001), with a projected separation obtained from the radio data of 20.08 arcsec (~13.3 kpc). Analysis of the optical spectra reveals a faint narrow extended emission from H α and [O III] amidst the two AGN, supporting evidence for an ongoing merger. IRAS 05589 + 2828 is a double component narrow emission-line AGN, with complex broad Balmer emission line profiles that clearly show a strong red-peaklet with a velocity shift of ~3500 km s-1. The black hole mass estimates of IRAS 05589 + 2828 and 2MASX J06021107 + 2828382 are log M $\rm _{BH}$ = 8.59 ± 0.14 (M⊙) and log M$\rm _{BH}$ = 8.21 ± 0.2 (M⊙), respectively. In the X-ray bands, IRAS 05589 + 2828 is compatible with a Type 1 object, showing both spectral and flux variability. Chandra data of 2MASX J06021107 + 2828382 allowed us to measure a high hardness ratio in this source, providing evidence for a Type 2 AGN. The 22 GHz image obtained with the Karl G. Jansky Very Large Array has revealed that both AGN are compact radio objects with spectral indices -0.26 ± 0.03 and -0.70 ± 0.11, confirming for the first time its dual AGN nature in the radio bands.