Task-oriented communication presents a promising approach to improve the communication efficiency of edge inference systems by optimizing learning-based modules to extract and transmit relevant task information. However, real-time applications face practical challenges, such as incomplete coverage and potential malfunctions of edge servers. This situation necessitates cross-model communication between different inference systems, enabling edge devices from one service provider to collaborate effectively with edge servers from another. Independent optimization of diverse edge systems often leads to incoherent feature spaces, which hinders the cross-model inference for existing task-oriented communication. To facilitate and achieve effective cross-model task-oriented communication, this study introduces a novel framework that utilizes shared anchor data across diverse systems. This approach addresses the challenge of feature alignment in both server-based and on-device scenarios. In particular, by leveraging the linear invariance of visual features, we propose efficient server-based feature alignment techniques to estimate linear transformations using encoded anchor data features. For on-device alignment, we exploit the angle-preserving nature of visual features and propose to encode relative representations with anchor data to streamline cross-model communication without additional alignment procedures during the inference. The experimental results on computer vision benchmarks demonstrate the superior performance of the proposed feature alignment approaches in cross-model task-oriented communications. The runtime and computation overhead analysis further confirm the effectiveness of the proposed feature alignment approaches in real-time applications.