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We present Multi-Unit Spectroscopic Explorer (MUSE) integral-field spectroscopy of ESO 253$-$G003, which hosts a known Active Galactic Nucleus (AGN) and the periodic nuclear transient ASASSN-14ko, observed as part of the All-weather MUse Supernova Integral-field of Nearby Galaxies (AMUSING) survey. The MUSE observations reveal that the inner region hosts two AGN separated by $1.4\pm0.1~\rm{kpc}$ ($\approx 1.\!\!^{\prime\prime}7$). The brighter nucleus has asymmetric broad, permitted emission-line profiles and is associated with the archival AGN designation. The fainter nucleus does not have a broad emission-line component but exhibits other AGN characteristics, including $v_{\rm{FWHM}}\approx 700~\rm{km}~\rm{s}^{-1}$ forbidden line emission, $\log_{10}(\rm{[OIII]}/\rm{H}β) \approx 1.1$, and high excitation potential emission lines such as [Fe$~$VII]$~\lambda6086$ and He$~$II$~\lambda4686$. The host galaxy exhibits a disturbed morphology with large kpc-scale tidal features, potential outflows from both nuclei, and a likely superbubble. A circular relativistic disk model cannot reproduce the asymmetric broad emission-line profiles in the brighter nucleus, but two non-axisymmetric disk models provide good fits to the broad emission-line profiles: an elliptical disk model and a circular disk + spiral arm model. Implications for the periodic nuclear transient ASASSN-14ko are discussed.