Federated learning is a distributed learning paradigm in which clients collaboratively train a shared model through local computation and server-side aggregation of model updates, without sharing raw data. However, under non-independent and identically distributed data settings, the aggregation performance of conventional FL often degrades significantly. Although clustered FL is an effective approach to mitigating data heterogeneity, existing methods still exhibit several limitations, including high computational overhead from full-model parameter clustering, insufficient characterization of layer-wise difference under monolithic whole-model clustering and aggregation, and inadequate information utilization caused by hard clustering. To address these issues, we proposed a new layer-wise fuzzy clustered FL paradigm that uses the scaling factors of batch normalization layers as local data representations. Low-dimensional BN scaling factors were used to replace full-model parameters for lightweight clustering. Then, we introduced a layer-wise independent clustering and aggregation mechanism that performs client clustering and model aggregation separately for different network layers to better capture layer-specific heterogeneity. Finally, fuzzy C-means (FCM) was applied at each layer to enable fuzzy clustering, allowing each client to belong to multiple clusters and enabling membership-weighted aggregation for improved information utilization. Extensive experiments on the MNIST, EMNIST, CIFAR-10, and CIFAR-100 datasets with SIMPLE-CNN and VGG16 demonstrated the effectiveness of the proposed approach over several state-of-the-art clustered FL baselines, with up to 7.10% gains in test accuracy over the strongest baseline. Further ablation studies showed that both the proposed fuzzy clustering mechanism and layer-wise aggregation strategy provide significant benefits, and their combination achieved the best overall performance.