Recent scanning tunneling microscopy experiments on electron transport through iron(II) phthalocyanine (FePc) molecules adsorbed on the Au(111) surface have revealed that the measured Kondo conductance signature depends strongly on the specific adsorption site. To understand the physical origin of experimental observations, particularly the variation of Kondo features with the molecular adsorption site, we employ a combined density functional theory (DFT) and hierarchical equations of motion (HEOM) approach to investigate the electronic structure and Kondo correlation in FePc/Au(111) composite systems. The calculation results indicate that, for the on-top adsorption configuration, the two degenerate spin-unpaired orbitals on the Fe center are coupled indirectly through substrate band states, leading to the Fano-like antiresonance line shape in the spectra, while for the bridge adsorption configuration, the environment-induced couplings are largely suppressed because of the two different spin-unpaired orbitals. Therefore, our work suggests that the environment-induced coupling as an essential physical factor could greatly influence the Fano-Kondo features in magnetic molecule/metal composites, and the crucial role of local orbital degeneracy and symmetry is discovered. These findings provide important insights into the electron correlation effects in complex solid-state systems. The usefulness and practicality of the combined DFT+HEOM method is also highlighted.