It really depends on how similar is the work that the different teams are doing; and also how large is each team and the volume of work they handle on an ongoing basis.
In most implementations that we have seen, most teams (not temporary project teams but long-running functional teams such as Dev, test or Design or Documentation or marketing) prefer to have their own boards for two reasons - one, they can have their own customized workflow(s) for the different type or priority of work they handle; and two, for better visibility, control of flow - and measurement of flow metrics such as CFD, Cycle Time and Flow Efficiency - the typical measures most teams implementing Kanban are interested in. If you have a Kanban board where multiple teams are working, you need to have the capability to let them define swim lanes with their own specific workflows - which I don't believe JIRA supports.
So, ideally, and depending on the response to the questions I asked earlier about the nature of your teams, each team should have their own Kanban boards - and the manager could have a "department-level" board where they can track the status of the work by teams based on some card-hierarchy that links cards on the manager's board to cards on the team boards. If that is not possible, then each team should at least have their own swim lane on a common board so that visualization, tracking, and measurement of metrics can be done cleanly.