  We analytically study the dynamics of a solitonic vortex (SV) in a superfluid
confined in a non-axisymmetric harmonic trap. The study provides a framework
for analyzing the role of the trap anisotropy in the oscillation of SVs
observed in recent experiments on atomic Bose and Fermi superfluids. The
emergence of common and statistics-dependent features is traced in a unified
approach to both types of fluid. Our description, built in the hydrodynamic
formalism, is based on a Lagragian approach which incorporates the vortex
location as dynamical parameters of a variational ansatz. Previous operative
Hamiltonian pictures are recovered through a canonically traced procedure. Our
results improve the understanding of the experimental findings. Some of the
observed features are shown to be specific to the tri-axial anisotropy of the
trap. In particular, we characterize the nontrivial dependence of the
oscillation frequency on the trapping transversal to the vortical line. The
study reveals also the crucial role played by the nonlinear character of the
dynamics in the observed oscillation: for the considered experimental
conditions, the frequency, and, in turn, the effective inertial mass of the
vortex, are found to significantly depend on the amplitude of the generated
motion. It is also uncovered how the coupling with collective modes of the
fluid induces a non-negligible shift in the oscillation frequency. The
appearance of fine-structure features in the SV trajectory is predicted.
