Yes it's possible, a simple post-function (script or plugin) would do. However, there are many ways to assign a ticket to someone in JIRA so to minimize the impact, I suggest that the "Assignee" can only be changed during a transition (remove it the native Assign button in JIRA, remove the Assignee in the "Edit Screen").
If you don't want to do that, you will have to create your own "Issue Updated listener" so that whenever the assignee is changed, you will update the label everytime.
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