A novel 3D-printed clamping interface was designed to address challenges associated with the tensile testing of soft biological tissues, particularly specimen slippage and failure at the grips. To improve specimen adherence, four contact patterns, based on retrograde teeth, serrated, atraumatic wavy teeth, and flower patterns, were added to the interface surface. A smooth transition was considered to diminish the likelihood of transverse cutting of specimens. The 3D-printed clamping interface was produced using additive manufacturing. We performed tensile tests on porcine skin specimens considering the original serrated jaw faces (reference condition), the jaw faces with sandpaper, and the 3D-printed clamping interface with and without contact patterns. The maximum force supported by the specimens (before slippage or failure), for each test condition, was compared using statistical analysis (statistical level of p < 0.05). Compared to the reference condition (148.50 ± 31.71 N), we observed significant improvements for the 3D-printed clamping interface with the retrograde teeth (247.41 ± 31.17 N, p ≤ 0.001) and flower (220.40 ± 19.86 N, p = 0.004) contact patterns. In the reference condition, failure mostly occurred at the grips. The use of the 3D-printed clamping interface reduced the spreading of the fibers, promoting failure within the gauge section. Additionally, we observed a reduction in tissue damage at the grips for the flower and atraumatic wavy teeth conditions. In conclusion, the proposed 3D-printed clamping interface significantly improved the adherence of the porcine skin specimens while promoting failure within the gauge section. This approach can be easily customized to the available grips, has a low-cost and fast production, and uses easily accessible technology.