Beyond diagonal reconfigurable intelligent surface (BD-RIS) is a family of RIS architectures more flexible than conventional RIS. While BD-RIS has been primarily analyzed assuming uni-polarized systems, modern wireless deployments are dual-polarized. To address this gap, this paper investigates the fundamental limits of dual-polarized BD-RIS-aided systems. We derive the scaling laws governing the performance of BD-RIS and the Pareto frontier of the trade-off between performance and circuit complexity enabled by BD-RIS. Theoretical results show that the group-connected RIS with group size 2 provides remarkable gains over conventional RIS in both Rayleigh and line-of-sight (LoS) channels, while maintaining a reduced circuit complexity.