This paper investigates the limits to which a passive Reconfigurable Intelligent Surface (RIS) can reshape a point-to-point Multiple-Input Multiple-Output (MIMO) channel in terms of singular values and their functions (e.g., achievable rate and harvestable power) for improved wireless performance. We depart from the Diagonal (D) scattering model and adopt a Beyond-Diagonal (BD) model that exploits element-wise connections for passive signal amplitude and phase manipulation. Specifically, analytical tight bounds are derived under typical RIS deployment scenarios to unveil the channel shaping potentials of BD-RIS regarding communication Degrees of Freedom (DoF), singular value spread, power gain, and capacity. An efficient numerical method is then proposed to optimize BD-RIS for any locally Lipschitz function of channel singular values, and showcased to characterize the achievable singular value region. As a side product, we tackle BD-RIS-aided MIMO rate maximization problem by a local-optimal Alternating Optimization (AO) approach and a low-complexity shaping approach. Results show that BD-RIS significantly improves the dynamic range of channel singular values and the tradeoff in manipulating them, thus offering enhanced data rate, harvestable power, and physical-layer security. These advantages become more pronounced when the number of RIS elements, group size, or MIMO dimensions increase. Of particular interest, BD-RIS is shown to activate multi-stream transmission and achieve the asymptotic DoF at much lower transmit power than D-RIS thanks to its proficiency in channel shaping.