Passive acoustic monitoring enables large-scale biodiversity assessment, but reliable classification of bioacoustic sounds requires not only high accuracy but also well-calibrated uncertainty estimates to ground decision-making. In bioacoustics, calibration is challenged by overlapping vocalisations, long-tailed species distributions, and distribution shifts between training and deployment data. The calibration of multi-label deep learning classifiers within the domain of bioacoustics has not yet been assessed. We systematically benchmark the calibration of four state-of-the-art multi-label bird sound classifiers on the BirdSet benchmark, evaluating both global, per-dataset and per-class calibration using threshold-free calibration metrics (ECE, MCS) alongside discrimination metrics (cmAP). Model calibration varies significantly across datasets and classes. While Perch v2 and ConvNeXt$_{BS}$ show better global calibration, results vary between datasets. Both models indicate consistent underconfidence, while AudioProtoPNet and BirdMAE are mostly overconfident. Surprisingly, calibration seems to be better for less frequent classes. Using simple post hoc calibration methods we demonstrate a straightforward way to improve calibration. A small labelled calibration set is sufficient to significantly improve calibration with Platt scaling, while global calibration parameters suffer from dataset variability. Our findings highlight the importance of evaluating and improving uncertainty calibration in bioacoustic classifiers.