I can’t remember where I got this screenshot, however, it is a good starting place for team structure.
I use teams within our company from a CEO viewpoint to keep similar projects grouped together. It would be your preference to keep all of the Product Department projects in one team, or make separate teams for each division. I assume the same people within each division would be working on the same projects. You could name them: Product - Product, Product - Business Analyses and Product - Tech Support to keep them alphabetically grouped in your master Team list. When it comes to viewing the projects in the Kanban view, you can customize the column headings in the Board view, however, I do think those become the Section headers in the list view. You could use a customized field to track statuses with a drop-down choice of: Todo, in progress, closed, etc. Then a sort in your list view would display current status. I would recommend another custom field for Priority, choosing High, Medium, Low. Then you can sort the priority column by those and bring the high priority to the top.