What is this device on my shower head?
The only thing that comes to my mind is a filter, probably activated charcoal, to remove residual chlorine. I doubt that it is necessary with the normal public water system. One sets the month when a new filter is installed as a reminder to change on some schedule.

Such a filter must be changed on some schedule because over time a biofilm will build up on the filter cartridge.

Edit

The original poster located this unit https://www.homedepot.com/p/Sprite-Showers-Universal-Shower-Filter-in-Chrome-SL2-CM-R/202386562