Deep learning (DL) has recently attracted increasing interest to improve object type classification for automotive radar.In addition to high accuracy, it is crucial for decision making in autonomous vehicles to evaluate the reliability of the predictions; however, decisions of DL networks are non-transparent. Current DL research has investigated how uncertainties of predictions can be quantified, and in this article, we evaluate the potential of these methods for safe, automotive radar perception. In particular we evaluate how uncertainty quantification can support radar perception under (1) domain shift, (2) corruptions of input signals, and (3) in the presence of unknown objects. We find that in agreement with phenomena observed in the literature,deep radar classifiers are overly confident, even in their wrong predictions. This raises concerns about the use of the confidence values for decision making under uncertainty, as the model fails to notify when it cannot handle an unknown situation. Accurate confidence values would allow optimal integration of multiple information sources, e.g. via sensor fusion. We show that by applying state-of-the-art post-hoc uncertainty calibration, the quality of confidence measures can be significantly improved,thereby partially resolving the over-confidence problem. Our investigation shows that further research into training and calibrating DL networks is necessary and offers great potential for safe automotive object classification with radar sensors.