Context. About 30% – 40% of the baryons in the local Universe remain unobserved. Many of these "missing" baryons are expected to reside in the warm-hot intergalactic medium (WHIM) of the cosmic web filaments that connect clusters of galaxies. SRG/eROSITA performance verification (PV) observations covered about 15 square degrees of the A3391/95 system and have revealed a ~15 Mpc continuous soft emission connecting several galaxy groups and clusters. Aims. We aim to characterize the gas properties in the cluster outskirts (R₅₀₀ < r < R₂₀₀) and in the detected inter-cluster filaments (> R₂₀₀) and to compare them to predictions. Methods. We performed X-ray image and spectral analyses using the eROSITA PV data in order to assess the gas morphology and properties in the outskirts and the filaments in the directions of the previously detected Northern and Southern Filament of the A3391/95 system. We constructed surface brightness profiles using particle-induced background-subtracted, exposure- and Galactic absorption-corrected eROSITA products in the soft band (0.3–2.0 keV). We constrained the temperatures, metallicities, and electron densities through X-ray spectral fitting and compared them with the expected properties of the WHIM. We took particular care of the foreground. Results. In the filament-facing outskirts of A3391 and the Northern Clump, we find higher temperatures than typical cluster outskirts profiles, with a significance of between 1.6 and 2.8σ, suggesting heating due to their connections with the filaments. We confirm surface brightness excess in the profiles of the Northern, Eastern, and Southern Filaments. From spectral analysis, we detect hot gas of 0.96_−0.14^+0.17 keV and 1.09_−0.06^+0.09 for the Northern and Southern Filament, respectively, which are close to the upper WHIM temperature limit. The filament metallicities are below 10% solar metallicity and the electron densities are ranging between 2.6 and 6.3 × 10⁻⁵ cm⁻³. The characteristic properties of the Little Southern Clump (LSC), which is located at a distance of ~1.5R₂₀₀ from A3395S in the Southern Filament, suggest that it is a small galaxy group. Excluding the LSC from the analysis of the Southern Filament does not significantly change the temperature or metallicity of the gas, but it decreases the gas density by 30%. This shows the importance of taking into account any clumps in order to avoid overestimation of the gas measurement in the outskirts and filament regions. Conclusions. We present measurements of morphology, temperature, metallicity, and density of individual warm-hot filaments. The electron densities of the filaments are consistent with the WHIM properties as predicted by cosmological simulations, but the temperatures are higher. As both filaments are short (1.8 and 2.7 Mpc) and located in a denser environment, stronger gravitational heating may be responsible for this temperature enhancement. The metallicities are low, but still within the expected range from the simulations.