In Brand Embassy there are two types of Assignees.
The Inbox Assignee is the agent who currently has the case in their Inbox. That means that if you have a tab open with a case you are working on you are the inbox assignee. Once you finish the work on the case and hit the close tab button, you will be asked if you are sure you want to be unassigned and once you confirm, you are no longer the inbox assignee of that case.
If you are the last agent to interact within that case you are the Owner Assignee. Therefore you will see the metrics generated for that case in the reports. In case the customer comes back within the same and another agent interacts, they will see the metrics in their reports (agent performance).
You can see the different assignees in the Case search in the CRM.