Recent advances in diffusion-based lip-syncing generative models have demonstrated their ability to produce highly synchronized talking face videos for visual dubbing. Although these models excel at lip synchronization, they often struggle to maintain fine-grained control over facial details in generated images. In this work, we identify "lip averaging" phenomenon where the model fails to preserve subtle facial details when dubbing unseen in-the-wild videos. This issue arises because the commonly used UNet backbone primarily integrates audio features into visual representations in the latent space via cross-attention mechanisms and multi-scale fusion, but it struggles to retain fine-grained lip details in the generated faces. To address this issue, we propose UnAvgLip, which extracts identity embeddings from reference videos to generate highly faithful facial sequences while maintaining accurate lip synchronization. Specifically, our method comprises two primary components: (1) an Identity Perceiver module that encodes facial embeddings to align with conditioned audio features; and (2) an ID-CrossAttn module that injects facial embeddings into the generation process, enhancing model's capability of identity retention. Extensive experiments demonstrate that, at a modest training and inference cost, UnAvgLip effectively mitigates the "averaging" phenomenon in lip inpainting, significantly preserving unique facial characteristics while maintaining precise lip synchronization. Compared with the original approach, our method demonstrates significant improvements of 5% on the identity consistency metric and 2% on the SSIM metric across two benchmark datasets (HDTF and LRW).