HESS J1507622 is an unusual TeV gamma-ray source in the H.E.S.S. Galactic Plane Survey, distinguished by its off-plane location () and compact angular extent (). Its location and morphology challenge conventional pulsar wind nebula and supernova remnant interpretations, and the absence of a detected shell or associated pulsar leaves its origin unresolved. Here, we propose that HESS J1507622 is a young Galactic kilonova remnant with gamma-ray emission of leptonic origin. In this scenario, the source's off-plane location is naturally explained by the natal kick imparted to the progenitor binary neutron star system. We demonstrate that a leptonic kilonova remnant model can successfully reproduce the observed GeV-TeV flux via inverse-Compton scattering from shock-accelerated electrons, while the corresponding synchrotron emission remains below current X-ray and radio upper limits. Based on the source's energetics and angular size, we infer a distance of 3.8-14.3 kpc and an age of 0.2-3.0 kyr. Importantly, our analysis favors a kilonova remnant younger than 1 kyr. The lack of a historical naked-eye record can be explained by the transient's rapid fading, far-southern declination, and potential line-of-sight extinction. We further assess the detectability of unique multi-wavelength signatures that could confirm this hypothesis. We find promising prospects for detecting the unique MeV gamma rays from the decay of -process radioisotopes using future observatories, alongside complementary low-energy signatures from thermal dust emission and a possible optical light echo. These distinct signals offer definitive tests of the kilonova remnant interpretation and motivate targeted observations with current and future facilities.