Many advanced driver assistance schemes or autonomous vehicle controllers are based on a motion model of the vehicle behavior, i.e., a function predicting how the vehicle will react to a given control input. Data-driven models, based on experimental or simulated data, are very useful, especially for vehicles difficult to model analytically, for instance, ground vehicles for which the ground-tire interaction is hard to model from first principles. However, learning schemes are limited by the difficulty of collecting large amounts of experimental data or having to rely on high-fidelity simulations. This paper explores the potential of an approach that uses dimensionless numbers based on Buckingham's $π$ theorem to improve the efficiency of data for learning models, with the goal of facilitating knowledge sharing between similar systems. A case study using car-like vehicles compares traditional and dimensionless models on simulated and experimental data to validate the benefits of the new dimensionless learning approach. Prediction accuracy improvements with the dimensionless scheme when using a shared database, that is, predicting the motion of a vehicle based on data from various different vehicles was found to be 480\% more accurate for predicting a simple no-slip maneuver based on simulated data and 11\% more accurate to predict a highly dynamic braking maneuver based on experimental data. A modified physics-informed learning scheme with hand-crafted dimensionless features was also shown to increase the improvement to precision gains of 917\% and 28\% respectively. A comparative study also shows that using Buckingham's $π$ theorem is a much more effective preprocessing step for this task than principal component analysis (PCA) or simply normalizing the data.