Semantic communication can significantly improve bandwidth utilization in wireless systems by exploiting the meaning behind raw data. However, the advancements achieved through semantic communication are closely dependent on the development of deep learning (DL) models for joint source-channel coding (JSCC) encoder/decoder techniques, which require a large amount of data for training. To address this data-intensive nature of DL models, federated learning (FL) has been proposed to train a model in a distributed manner, where the server broadcasts the DL model to clients in the network for training with their local data. However, the conventional FL approaches suffer from catastrophic degradation when client data are from different domains. In contrast, in this paper, a novel FL framework is proposed to address this domain shift by constructing the global representation, which aligns with the local features of the clients to preserve the semantics of different data domains. In addition, the dominance problem of client domains with a large number of samples is identified and, then, addressed with a domain-aware aggregation approach. This work is the first to consider the domain shift in training the semantic communication system for the image reconstruction task. Finally, simulation results demonstrate that the proposed approach outperforms the model-contrastive FL (MOON) framework by 0.5 for PSNR values under three domains at an SNR of 1 dB, and this gap continues to widen as the channel quality improves.