Biochar aging is a key factor leading to the decline of biochar stability and the release of endogenous pollutants. This study investigated the effects of five artificial and simulated aging processes on the surface properties and endogenous copper (Cu) and zinc (Zn) leachability of swine manure biochar and its composite with alkali-fused fly ash. Aging obviously reduced carbon (C) content on the surface of swine manure biochar and increased oxygen (O) content. Among all the aging treatments, high-temperature aging had the greatest effect on C content. Following the aging treatments, the C–C bond contents on the surfaces of swine manure biochar decreased significantly, whereas the C–O bonds increased significantly; however, there were less changes in the amounts of C–C and C–O bonds on the surfaces of modified biochar than on swine manure biochar. Aging significantly enhanced the leaching toxicity of Cu and Zn, and Zn availability and bioaccessibility in swine manure biochar and modified biochar. However, it minimized Cu availability and bioaccessibility, especially under high-temperature aging. Greater amounts of Zn than Cu were extracted from swine manure biochar and modified biochar. However, under all the aging treatments, the leaching toxicity, availability, and bioaccessibility of Cu and Zn in modified biochar were significantly lower than in swine manure biochar. This implies that modified biochar application poses lower environmental risks than swine manure biochar.