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 that the case has been assigned to (through the Give me more button or automatically by the system), you are the Owner Assignee. You will see the metrics generated for this case in your reports. In case the customer replies again on the same case and the next agent gets this case assigned to them, they will see the metrics in their reports (Agent Performance).
You can see both different assignees in the Case search in the CRM.