Context. The late evolutionary stages of low- and intermediate-mass stars are characterised by mass loss through a dust-driven stellar wind. Recent observations reveal complex structures within these winds, which are believed to be formed primarily via an interaction with a companion. How these complexities arise, and which structures are formed in which type of systems, is still poorly understood. Particularly, there is a lack of studies investigating the structure formation in eccentric systems. Aims. We aim to improve our understanding of the wind morphology of eccentric asymptotic giant branch (AGB) binary systems by investigating the mechanism responsible for the different small-scale structures and global morphologies that arise in a polytropic wind with different velocities. Methods. Using the smoothed particle hydrodynamics (SPH) code PHANTOM, we generated nine different high-resolution, 3D simulations of an AGB star with a solar-mass companion with various wind velocity and eccentricity combinations. The models assume a polytropic gas, with no additional cooling. Results. Compared to the zero-eccentricity situation, we find that for low eccentricities, for the case of a high wind velocity, and hence limited interaction between the wind and the companion, the standard two-edged spiral structure that dominates the shape of the wind in the orbital plane is only minimally affected. When the wind speed is lower, strong compression of the wind material by the companion occurs, causing a high-pressure region around the companion which shapes the wind into an irregular spiral. In extreme cases, with low wind velocity and high eccentricity, these instabilities grow to such proportion that they cause high-speed ejections of matter along the orbital plane, shaping the wind into a highly irregular morphology. In more eccentric orbits, the amplitude of the phase-dependent wind-companion interaction increases significantly, introducing additional complexities that make the outbursts even more energetic, leading in some cases to high-speed polar flows of matter. Further, the orbital motion of the stars tends to flatten the global density distribution of the models with no instabilities. We distinguish global flattening from an equatorial density enhancement, the latter being formed by a strong gravitational interaction of the companion with the wind particles. We classify the resulting morphologies according to these new definitions, and find that (i) all low-velocity models have an equatorial density enhancement and (ii), in general, the flattening increases for decreasing wind velocity, until the low wind velocity results in high-energy outflows that clear away the flattening. Conclusions. We conclude that for models with a high wind velocity, the short interaction with the companion results in a regular spiral morphology, which is flattened. In the case of a lower wind velocity, the stronger interaction results in the formation of a high-energy region and bow-shock structure that can shape the wind into an irregular morphology if instabilities arise. High-eccentricity models show a complex, phase-dependent interaction leading to wind structures that are irregular in three dimensions. However, the significant interaction with the companion compresses matter into an equatorial density enhancement, irrespective of eccentricity.