From the perspective of joint source-channel coding (JSCC), there has been significant research on utilizing semantic communication, which inherently possesses analog characteristics, within digital device environments. However, a single-model approach that operates modulation-agnostically across various digital modulation orders has not yet been established. This article presents the first attempt at such an approach by proposing a universal joint source-channel coding (uJSCC) system that utilizes a single-model encoder-decoder pair and trained vector quantization (VQ) codebooks. To support various modulation orders within a single model, the operation of every neural network (NN)-based module in the uJSCC system requires the selection of modulation orders according to signal-to-noise ratio (SNR) boundaries. To address the challenge of unequal output statistics from shared parameters across NN layers, we integrate multiple batch normalization (BN) layers, selected based on modulation order, after each NN layer. This integration occurs with minimal impact on the overall model size. Through a comprehensive series of experiments, we validate that the modulation-agnostic semantic communication framework demonstrates superiority over existing digital semantic communication approaches in terms of model complexity, communication efficiency, and task effectiveness.