The Host Galaxy of the Recoiling Black Hole Candidate in 3C 186: An Old Major Merger Remnant at the Center of a z=1 Cluster

3C 186, a radio-loud quasar at z = 1.0685, was previously reported to have both velocity and spatial offsets from its host galaxy, and has been considered as a promising candidate for a gravitational wave recoiling black hole triggered by a black hole merger. Another possible scenario is that 3C 186 is in an ongoing galaxy merger, exhibiting a temporary displacement. In this study, we present analyses of new deep images from the Hubble Space Telescope WFC3-IR and Advanced Camera for Surveys, aiming to characterize the host galaxy and test this alternative scenario. We carefully measure the light-weighted center of the host and reveal a significant spatial offset from the quasar core (11.1 +/- 0.1 kpc). The direction of the confirmed offset aligns almost perpendicularly to the radio jet. We do not find evidence of a recent merger, such as a young starburst in disturbed outskirts, but only marginal light concentration in F160W at similar to 30 kpc. The host consists of mature (greater than or similar to 200 Myr) stellar populations and one compact star-forming region. We compare with hydrodynamical simulations and find that those observed features are consistently seen in late-stage merger remnants. Taken together, those pieces of evidence indicate that the system is not an ongoing/young merger remnant, suggesting that the recoiling black hole scenario is still a plausible explanation for the puzzling nature of 3C 186.