Remoras are a family of fishes that can attach to other swimming organisms via an adhesive disc evolved from dorsal fin elements. However, the factors driving the evolution of remora disc morphology are poorly understood. It is not possible to link selective pressure for attachment to a specific host surface because all known hosts evolved before remoras themselves. Fortunately, the fundamental physics of suction and friction are mechanically conserved. Therefore, a morphologically relevant bioinspired model can be used to examine performance of hypothetical evolutionary intermediates. Using a bioinspired remora disc, we experimentally investigated the performance of increased lamellar number on shear adhesion. Herein, we translated fundamental biological principles into engineering design rules and show that a passive model system can autonomously achieve adhesive forces measured in live remoras in any environment. Our experimental results show that an increase in lamellar number resulted in an increase in shear adhesive performance, supporting the phylogenetic trend observed in extant remoras. The greatest pull-off forces measured for our model were on surface roughness on the order of shark skin and exceeded those measured for live remoras attached to shark skin by almost 60%. Overall, relative to fossil remoras and their closest ancestor, extant remoras exhibit a morphology indicative of selection for enhanced shear adhesive performance.