Biohybrid systems in which robotic lures interact with animals have become compelling tools for probing and identifying the mechanisms underlying collective animal behavior. One key challenge lies in the transfer of social interaction models from simulations to reality, using robotics to validate the modeling hypotheses. This challenge arises in bridging what we term the 'biomimicry gap', which is caused by imperfect robotic replicas, communication cues and physics constraints not incorporated in the simulations, that may elicit unrealistic behavioral responses in animals. In this work, we used a biomimetic lure of a rummy-nose tetra fish (Hemigrammus rhodostomus) and a neural network (NN) model for generating biomimetic social interactions. Through experiments with a biohybrid pair comprising a fish and the robotic lure, a pair of real fish, and simulations of pairs of fish, we demonstrate that our biohybrid system generates social interactions mirroring those of genuine fish pairs. Our analyses highlight that: 1) the lure and NN maintain minimal deviation in real-world interactions compared to simulations and fish-only experiments, 2) our NN controls the robot efficiently in real-time, and 3) a comprehensive validation is crucial to bridge the biomimicry gap, ensuring realistic biohybrid systems.