This work presents a comparative study on the mechanical homogenisation of Triply Periodic Minimal Surface (TPMS) lattice structures in the linear elastic regime, which have attracted significant interest for their unique ability to combine lightweight design with tailored properties. The study investigates the effective mechanical behaviour of Representative Unit Cells (RUCs) generated using the open-source Python tool Microgen. Two homogenisation strategies are considered: (i) Finite Element (FE)-based homogenisation carried out in Abaqus, and (ii) the Mechanics of Structure Genome (MSG), a unified theory for multi-scale constitutive modelling, implemented in an in-house software tool. The comparison encompasses multiple TPMS topologies, including well-studied cases used for validation, namely gyroid and diamond, as well as less-explored ones, such as PMY and F-Rhombic Dodecahedron, to provide new insights. RUCs are analysed across relative densities ranging from 10 to 50%. Equivalent linear elastic properties (Young’s moduli, shear moduli, and Poisson’s ratios) are derived and compared to assess the consistency, accuracy, and computational efficiency of the two approaches. The results show that both methods yield effective properties with less than 1% difference between them, and less than 5% deviation from experimental data reported in the literature for the effective Young’s modulus. Furthermore, the anisotropy of each TPMS topology across the range of relative densities is examined through the directional distribution of Young’s moduli. The outcomes are expected to clarify the strengths and limitations of FE versus MSG in capturing the effective behaviour of architected cellular solids, thus supporting the selection of homogenisation strategies for the design of lattice-based lightweight structures.