Skyrmions are particlelike vortices of magnetization with nontrivial topology, which are usually stabilized by Dzyaloshinskii-Moriya interactions (DMI) in noncentrosymmetric bulk materials. Exceptions are centrosymmetric Gd- and Eu-based skyrmion-lattice (SL) hosts with zero DMI, where both the SL stabilization mechanisms and magnetic ground states remain controversial. We address these here by investigating both the static and dynamical spin properties of the centrosymmetric SL host Gd_{2}PdSi_{3} using muon spectroscopy. We find that spin fluctuations in the noncoplanar SL phase are highly anisotropic, implying that spin anisotropy plays a prominent role in stabilizing this phase. We also observe strongly anisotropic spin dynamics in the ground-state (IC-1) incommensurate magnetic phase of the material, indicating that it hosts a meronlike multi-q structure. In contrast, the higher-field, coplanar IC-2 phase is found to be single q with nearly isotropic spin dynamics.