  Skyrmions are particle-like vortices of magnetization with non-trivial
topology, which are usually stabilized by Dzyaloshinskii-Moriya interactions
(DMI) in noncentrosymmetric bulk materials. Exceptions are centrosymmetric Gd-
and Eu-based skyrmion-lattice (SkL) hosts with net-zero DMI, where both the SkL
stabilization mechanisms and magnetic ground states remain controversial. We
address these by investigating both static and dynamic spin properties of the
centrosymmetric SkL host Gd$_2$PdSi$_3$ using muon spectroscopy ($\mu$SR). We
find that spin fluctuations in its non-coplanar SkL 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 is a meron-like multi-$q$ structure. In contrast, the
higher-field, coplanar IC-2 phase is found to be single-$q$ with
nearly-isotropic spin dynamics.
