The elegant spin physics of Dirac electrons in topological insulators (TIs) have considerably endowed fertile tunability of magnetic/TI heterojunction performance with modified spin-orbit effect engineering. Signatures of proximate hybridization between magnetic states and topological surface states have been reported. However, the nature of the spin relaxation process in these systems remains elusive. Here, we unambiguously demonstrate anisotropic spin relaxation in a spin-orbit-hybridized $\mathrm{Fe}\text{/}{\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ system. We find a sixfold anisotropy of the Gilbert damping parameter with modulation of up to 33% in $\mathrm{Fe}\text{/}{\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ in the presence of a topological surface state, together with a sixfold magnetic anisotropy. We anticipate the presence of a spin interplay between the topological spin-orbit texture and magnetic orbital states would manifest an anisotropic Gilbert damping, which corroborates with the density functional theory calculations. It is further demonstrated by the spin Hanle effect indicative of anisotropic spin relaxation time ${\ensuremath{\tau}}_{s}$ in the adjacent topological layer, inversely scaling with the Gilbert damping factor ${\ensuremath{\alpha}}_{G}$. Our findings present an alternative scenario of the anisotropic spin transport process and offer insights into spin manipulation in spin-logic/memory devices utilizing proximity-hybridized Dirac electrons.