Topology optimization provides a powerful framework for designing lightweight and mechanically efficient structures across multiple length scales. This work presents a concurrent multiscale, multi-material topology optimization approach incorporating Triply Periodic Minimal Surface (TPMS) architected materials. The optimization problem minimizes structural compliance subject to multiple volume constraints using a gradient-based formulation. TPMS microstructures are described through the Fourier Series Function method, and their effective properties are incorporated into the macroscale analysis by homogenization. Material interpolation combines the Solid Isotropic Material with Penalization scheme with Discrete Material Optimization and employs a continuation strategy to progressively drive the design from a convex formulation toward a discrete, non-convex solution. An adaptation of the ZPR-BFGS update scheme enables the efficient treatment of multiple volume constraints, which may apply globally or locally to selected material phases and can be updated independently, sequentially, or in parallel. A voxel-based post-processing procedure is also introduced to generate geometrically consistent designs with smooth material transitions and controllable scale separation. Numerical examples demonstrate the effectiveness and flexibility of the proposed framework in distributing TPMS architectures within structures of arbitrary geometric complexity. [1] A. Nale and A. Chiozzi. Multiscale topology optimization with embedded TPMS architected materials. International Journal of Mechanical Sciences, Vol. 295, 110204, 2025. [2] E.D. Sanders, M.A. Aguiló and G.H. Paulino. Multi-material continuum topology optimization with arbitrary volume and mass constraints. Computer Methods in Applied Mechanics and Engineering, Vol. 380, pp. 798-823, 2018.