Unmanned aerial vehicles (UAVs) and UAV traffic management (UTM) have drawn attention for applications such as parcel delivery, aerial mapping, agriculture, and surveillance based on line-of-sight (LoS) links. UTM is essential to operate multiple fully autonomous UAVs safely beyond the visual line of sight (BVLoS) in the future dense UAV traffic environment. Various research and development teams globally take UTM initiatives and work on platform testing with different industrial partners. In the future, urban airspace will be congested with various types of autonomous aerial vehicles, thereby resulting in complex air-traffic management caused by communication issues. The UTM requires an efficient communication backbone to handle all airborne communication services. Existing cellular networks are suitable only for terrestrial communication and have limitations in supporting aerial communications. These issues motivate the investigation of an appropriate communication technology for advanced UTM systems. Thus, in this study, we present a future perspective of 6G-enabled UTM ecosystems in a very dense and urban air-traffic scenario focusing on non-terrestrial features, including aerial and satellite communication. We also introduce several urban airspace segmentation and discuss a strategic management framework for dynamic airspace traffic management and conflict-free UAV operations. The UTM enhances the adaptive use of the airspace by shaping the airspace with the overall aim of maximizing the capability and efficiency of the network. We also discuss the 6G multi-layer parameters i.e., space, air, and terrestrial, for safe and efficient urban air transportation in three-dimensional space. Moreover, we discuss the issues and challenges faced by future UTM systems and provide tentative solutions. We subsequently extend the vision of the UTM system and design an advanced and fully autonomous 6G-based UTM system.