Recent advances in deep learning have led to significant improvements in nuclei segmentation from histological images, particularly when labels of all classes are available simultaneously during training. However, in clinical practice, real-world scenarios require a model to perform well in an incremental learning setting, where we anticipate the model to achieve satisfactory performance on previously unseen data while effectively mitigating catastrophic forgetting of old classes. Most previous methods alleviate forgetting by distilling old class knowledge through prototypes; however, they fail to adequately capture fine-grained details to address the challenge of high class similarity, which is particularly severe in histological images. To overcome these limitations, we propose a novel incremental learning method for nuclei segmentation (we call it CiNuSeg), which is composed of two key innovative modules. First, we propose a new Anchor-driven Consistency Learning (ACL) module to construct multi-level class anchors within each sample to effectively capture fine structural and textural details of nuclei, thereby significantly mitigating forgetting. Second, we develop a Dual Region Regularization (DRR) module to suppress new class representations within old class regions while enhancing new class representations within new class regions, strengthening the model's ability to discriminate between different nuclei types and improving inter-class separability. We further introduce an Adaptive Temperature Tuning (ATT) strategy to dynamically balance model stability and plasticity. Extensive experiments conducted on benchmarking MoNuSAC and CoNSeP pathological datasets demonstrate the effectiveness of our method, consistently achieving better performance than SOTAs in different settings. Codes will be available upon publication.