Using 3D point clouds in odometry estimation in robotics often requires finding a set of correspondences between points in subsequent scans. While there are established methods for point clouds of sufficient quality, state-of-the-art still struggles when this quality drops. Thus, this paper presents a novel learning-based framework for predicting robust point correspondences between pairs of noisy, sparse and unstructured 3D point clouds from a light-weight, low-power, inexpensive, consumer-grade System-on-Chip (SoC) Frequency Modulated Continuous Wave (FMCW) radar sensor. Our network is based on the transformer architecture which allows leveraging the attention mechanism to discover pairs of points in consecutive scans with the greatest mutual affinity. The proposed network is trained in a self-supervised way using set-based multi-label classification cross-entropy loss, where the ground-truth set of matches is found by solving the Linear Sum Assignment (LSA) optimization problem, which avoids tedious hand annotation of the training data. Additionally, posing the loss calculation as multi-label classification permits supervising on point correspondences directly instead of on odometry error, which is not feasible for sparse and noisy data from the SoC radar we use. We evaluate our method with an open-source state-of-the-art Radar-Inertial Odometry (RIO) framework in real-world Unmanned Aerial Vehicle (UAV) flights and with the widely used public Coloradar dataset. Evaluation shows that the proposed method improves the position estimation accuracy by over 14 % and 19 % on average, respectively. The open source code and datasets can be found here: https://github.com/aau-cns/radar_transformer.