Federated learning (FL) enables edge devices to collaboratively train machine learning models, with model communication replacing direct data uploading. While over-the-air model aggregation improves communication efficiency, uploading models to an edge server over wireless networks can pose privacy risks. Differential privacy (DP) is a widely used quantitative technique to measure statistical data privacy in FL. Previous research has focused on over-the-air FL with a single-antenna server, leveraging communication noise to enhance user-level DP. This approach achieves the so-called "free DP" by controlling transmit power rather than introducing additional DP-preserving mechanisms at devices, such as adding artificial noise. In this paper, we study differentially private over-the-air FL over a multiple-input multiple-output (MIMO) fading channel. We show that FL model communication with a multiple-antenna server amplifies privacy leakage as the multiple-antenna server employs separate receive combining for model aggregation and information inference. Consequently, relying solely on communication noise, as done in the multiple-input single-output system, cannot meet high privacy requirements, and a device-side privacy-preserving mechanism is necessary for optimal DP design. We analyze the learning convergence and privacy loss of the studied FL system and propose a transceiver design algorithm based on alternating optimization. Numerical results demonstrate that the proposed method achieves a better privacy-learning trade-off compared to prior work.