Most learning-based lossless compressors are designed for a single modality, requiring separate models for multi-modal data and lacking flexibility. However, different modalities vary significantly in format and statistical properties, making it ineffective to use compressors that lack modality-specific adaptations. While multi-modal large language models (MLLMs) offer a potential solution for modality-unified compression, their excessive complexity hinders practical deployment. To address these challenges, we focus on the two most common modalities, image and text, and propose DualComp, the first unified and lightweight learning-based dual-modality lossless compressor. Built on a lightweight backbone, DualComp incorporates three key structural enhancements to handle modality heterogeneity: modality-unified tokenization, modality-switching contextual learning, and modality-routing mixture-of-experts. A reparameterization training strategy is also used to boost compression performance. DualComp integrates both modality-specific and shared parameters for efficient parameter utilization, enabling near real-time inference (200KB/s) on desktop CPUs. With much fewer parameters, DualComp achieves compression performance on par with the SOTA LLM-based methods for both text and image datasets. Its simplified single-modality variant surpasses the previous best image compressor on the Kodak dataset by about 9% using just 1.2% of the model size.