Most existing deep learning-based PET image denoising methods assume a fixed and known dose reduction factor (DRF) for low-dose PET images. However, these methods encounter significant performance degradation when the DRF varies beyond the assumed one in practical applications. To address the challenge posed by varied DRFs, several preliminary studies focus on the task of universal PET image denoising, aiming to train a universal model over low-dose data across DRFs. Nonetheless, these vanilla universal models often struggle with misaligned styles present in different DRF data, leading to the \textit{style elimination issue} with a significant over-smoothing effect. To deal with this issue, we innovatively introduce domain generalization to PET image denoising and propose a universal PET image denoising network (UniPET) to achieve high-quality PET image denoising across diverse DRFs. UniPET comprises two primary innovations: a style alignment network (SAN) and a region-aware learning strategy (RALS). Specifically, SAN utilizes style alignment techniques derived from domain generalization to align and recover styles across different DRFs, ensuring the model's generalizability across various DRFs while effectively preserving styles. Furthermore, to enhance style recovery, RALS distinguishes between flat and stylized regions, exclusively conducting adversarial learning on the latter, thereby more effectively guiding the model's focus towards learning stylized regions. It is demonstrated that our proposed UniPET can adaptively recover different DRF styles and achieve high-quality PET image denoising across DRFs. Comprehensive experiments show that UniPET exhibits comparable performance to individual DRF-specific models at specific DRFs and realizes state-of-the-art performance in universal PET image denoising quantitatively, perceptually, and clinically.