The ArcGIS Enterprise portal members with custom roles must be able to delete their own services when the role includes administrative privileges such as 'View all members’ and publisher privileges.
This issue is resolved in ArcGIS Enterprise 10.8.1 by installing both the Portal for ArcGIS 10.8.1 Custom Roles with Administrative Privileges Patch and the ArcGIS Server 10.8.1 Custom Roles with Administrative Privileges Patch. Refer to https://support.esri.com/en/download/7858 and https://support.esri.com/en/download/7860.