Space-air-ground-integrated network (SAGIN)-enabled multiconnectivity (MC) is emerging as a key enabler for next-generation networks, enabling users to simultaneously utilize multiple links across multi-layer non-terrestrial networks (NTN) and multi-radio access technology (multi-RAT) terrestrial networks (TN). However, the heterogeneity of TN and NTN introduces complex architectural challenges that complicate MC implementation. Specifically, the diversity of link types, spanning air-to-air, air-to-space, space-to-space, space-to-ground, and ground-to-ground communications, renders optimal resource allocation highly complex. Recent advancements in reinforcement learning (RL) and agentic artificial intelligence (AI) have shown remarkable effectiveness in optimal decision-making in complex and dynamic environments. In this paper, we review the current developments in SAGIN-enabled MC and outline the key challenges associated with its implementation. We further highlight the transformative potential of AI-driven approaches for resource optimization in a heterogeneous SAGIN environment. To this end, we present a case study on resource allocation optimization enabled by agentic RL for SAGIN-enabled MC involving diverse radio access technologies (RATs). Results show that learning-based methods can effectively handle complex scenarios and substantially enhance network performance in terms of latency and capacity while incurring a moderate increase in power consumption as an acceptable tradeoff. Finally, open research problems and future directions are presented to realize efficient SAGIN-enabled MC.