Federated continual learning (FCL) tackles scenarios of learning from continuously emerging task data across distributed clients, where the key challenge lies in addressing both temporal forgetting over time and spatial forgetting simultaneously. Recently, prompt-based FCL methods have shown advanced performance through task-wise prompt communication.In this study, we underscore that the existing prompt-based FCL methods are prone to class-wise knowledge coherence between prompts across clients. The class-wise knowledge coherence includes two aspects: (1) intra-class distribution gap across clients, which degrades the learned semantics across prompts, (2) inter-prompt class-wise relevance, which highlights cross-class knowledge confusion. During prompt communication, insufficient class-wise coherence exacerbates knowledge conflicts among new prompts and induces interference with old prompts, intensifying both spatial and temporal forgetting. To address these issues, we propose a novel Class-aware Client Knowledge Interaction (C${}^2$Prompt) method that explicitly enhances class-wise knowledge coherence during prompt communication. Specifically, a local class distribution compensation mechanism (LCDC) is introduced to reduce intra-class distribution disparities across clients, thereby reinforcing intra-class knowledge consistency. Additionally, a class-aware prompt aggregation scheme (CPA) is designed to alleviate inter-class knowledge confusion by selectively strengthening class-relevant knowledge aggregation. Extensive experiments on multiple FCL benchmarks demonstrate that C${}^2$Prompt achieves state-of-the-art performance. Our source code is available at https://github.com/zhoujiahuan1991/NeurIPS2025-C2Prompt