By leveraging the superposition property, over-the-air computation (OAC) of waveforms enables computations to be performed in an analog fashion in wireless environments, leading to faster computation, lower latency, and reduced energy consumption. In this paper, we investigate reconfigurable intelligent surface (RIS)-aided multiple-input-multiple-output (MIMO) OAC systems designed to emulate the fully-connected (FC) layer of a neural network (NN) via analog OAC, where the RIS and the transceivers are jointly adjusted to engineer the ambient wireless propagation environment to emulate the weights of the target FC layer. We refer to this novel computational paradigm as AirFC. We first study the case in which the precoder, combiner, and RIS phase shift matrices are jointly optimized to minimize the mismatch between the OAC system and the target FC layer. To solve this non-convex optimization problem, we propose a low-complexity alternating optimization algorithm, where semi-closed-form/closed-form solutions for all optimization variables are derived. Next, we consider training of the system parameters using two distinct learning strategies, namely centralized training and distributed training. In the centralized training approach, training is performed at either the transmitter or the receiver, whichever possesses the channel state information (CSI), and the trained parameters are provided to the other terminal. In the distributed training approach, the transmitter and receiver iteratively update their parameters through back and forth transmissions by leveraging channel reciprocity, thereby avoiding CSI acquisition and significantly reducing computational complexity. Subsequently, we extend our analysis to a multi-RIS scenario by exploiting its spatial diversity gain to enhance the system performance, i.e., classification accuracy. Simulation results show that the AirFC system realized by the RIS-aided MIMO configuration achieves satisfactory classification accuracy. Furthermore, it is shown that the multi-RIS system brings significant improvement in terms of the classification accuracy, especially in line-of-sight (LoS)-dominated wireless environments.