I know what it is and what it looked like but I cannot find a lot of information on its origin and use.
The swagger stick has been used in various different militaries and in it’s current form seems to have originated as a fashion item for off duty uniformed personnel. I can speak for the recent (up to 20 years ago) use of the swagger stick in the Australian Army. It is a symbol of authority carried either on parade or on duty, usually by Non Commissioned Officers (NCOs), Warrant Officers (other than the Regimental Sergeant Major who carries a Pace Stick) or junior commissioned officers. In the Australian Army the descriptor “swagger stick” is the same as “cane”.
It helps if you understand that, like a rifle, there is a drill for everything you do while holding or carrying a cane; marching, halting, turning, saluting, changing hands etc. so that while particular people may carry a cane, it isn’t very popular which makes it unusual and usually in Army units the personnel who carry a cane do so because in that unit, particular people are required to carry a cane by unit standing orders.
It varies with units but the common personnel who carry a cane are the duty sergeant (orderly sergeant) or in training establishments, the instructors or directing staff. That is, it provides immediate visibility from a distance that the person carrying it is exercising some additional authority, not just because of his rank but also because of the position or task he is currently performing.