Multi-class event streams arise in numerous real-world applications, where uncovering structured, interpretable inter-event relationships, together with accurate prediction, remains a central challenge. Existing neural point process models are highly expressive but encode event interactions in a black-box manner, preventing explicit discovery of structured dependencies. In this paper, we propose a structured neural marked point process (SNMPP) that achieves high modeling flexibility while enabling explicit event-wise and class-wise relationship discovery from data. Our model constructs a product-form neural influence kernel composed of a signed interaction network over event types and a delay-aware monotonic temporal network. This design enables explicit characterization of inter-class influence topology -- including excitation, inhibition, and neutrality -- while flexibly capturing diverse temporal decay patterns and potential influence delays. For efficient learning, we develop a stratified Monte Carlo estimator for stochastic training. Extensive experiments on synthetic and real-world benchmark datasets validate the ability of our approach to uncover structured relationships and deliver strong predictive performance.