Cooperative Intelligent Transportation Systems (C-ITS) enable communications between vehicles, road-side infrastructures, and road-users to improve users' safety and to efficiently manage traffic. Most, if not all, of the intelligent vehicles-to-everything (V2X) applications, often rely on continuous collection and sharing of sensitive information such as detailed location information which raises privacy concerns. In this light, a common approach to concealing the long-term identity of C-ITS vehicles is using multiple temporary identifiers, called pseudonyms. However, the legacy pseudonyms management approach is prone to linking attacks. The introduction of 5G network to V2X offers enhanced location accuracy, better clock synchronisation, improved modular service-based architecture, and enhanced security and privacy preservation controls. Motivated by the above enhancements, we study 5G-enabled pseudonyms for protecting vehicle identity privacy in C-ITS. We highlight the gaps in the current standards of pseudonyms management. We further provide recommendations regarding the pseudonyms management life-cycle.