Edge computing and network slicing might be considered as the main pillars of the upcoming 5G systems as they inject flexibility into the network management operations. While one prominent architectural framework for edge computing has been recently defined by the ETSI standard organization, namely multi-access edge computing (MEC), network slicing has reached its momentum by fostering interest in different standardization bodies and fora. To better understand how such distinct network slicing definitions impact the standardized MEC framework, ETSI has recently published a study on the matter. In this article, we first overview with a comprehensive analysis the different network slicing concepts and their relationship. Then we elaborate on the ETSI study to provide an integrated view of network slicing technology within the context of MEC. Finally, we report on the open challenges in the ETSI study and propose two solutions to evolve the current MEC framework toward end-to-end multi-slice support and efficient multi-tenant inter-slice communication in 5G deployments.