Millimeter-wave and terahertz communications face a fundamental challenge: overcoming severe path loss without sacrificing spectral efficiency. Pinching antenna systems (PASS) address this by bringing radiators physically close to users, yet existing frameworks treat the waveguide as a mere transmission line, overlooking its inherent multi-mode capabilities and the critical role of polarization. This paper develops the first polarization-aware, full-wave electromagnetic model for multi-mode PASS (MMPASS), capturing spatial radiation patterns, modal polarization states, and polarization matching efficiency from first principles. Leveraging this physically grounded model, we reveal fundamental trade-offs among waveguide attenuation, atmospheric absorption, and geometric spreading, yielding closed-form solutions for optimal PA placement and orientation in single-user scenarios. Extending to multi-user settings, we propose a modular optimization framework that integrates fractional programming with closed-form polarization updates, scaling gracefully to arbitrary numbers of waveguides, PAs, and users. Numerical results show that MMPASS achieves up to a 167% increase in spectral efficiency compared with single-mode PASS. Moreover, when comparing MMPASS with its polarization-ignorant counterpart, polarization awareness alone improves the sum rate by up to 23%. By bridging rigorous electromagnetic theory with scalable optimization, MMPASS establishes a physically complete and practically viable foundation for future high-frequency wireless networks.