Machine learning has the potential to improve the speed and reliability of radioisotope identification using gamma spectroscopy. However, meticulously labeling an experimental dataset for training is often prohibitively expensive, while training models purely on synthetic data is risky due to the domain gap between simulated and experimental measurements. In this research, we demonstrate that supervised domain adaptation can substantially improve the performance of radioisotope identification models by transferring knowledge between synthetic and experimental data domains. We consider two domain adaptation scenarios: (1) a simulation-to-simulation adaptation, where we perform multi-label proportion estimation using simulated high-purity germanium detectors, and (2) a simulation-to-experimental adaptation, where we perform multi-class, single-label classification using measured spectra from handheld lanthanum bromide (LaBr) and sodium iodide (NaI) detectors. We begin by pretraining a spectral classifier on synthetic data using a custom transformer-based neural network. After subsequent fine-tuning on just 64 labeled experimental spectra, we achieve a test accuracy of 96% in the sim-to-real scenario with a LaBr detector, far surpassing a synthetic-only baseline model (75%) and a model trained from scratch (80%) on the same 64 spectra. Furthermore, we demonstrate that domain-adapted models learn more human-interpretable features than experiment-only baseline models. Overall, our results highlight the potential for supervised domain adaptation techniques to bridge the sim-to-real gap in radioisotope identification, enabling the development of accurate and explainable classifiers even in real-world scenarios where access to experimental data is limited.