WallyO, Salt brine from the brine tank is required to regenerate softening capacity in a water softener.
The first tank not connected to the brine tank, is most likely a filter of some type.
You didn't state your water source. Carbon filtration media is often utilized to remove chlorine or Chloramine, either of which is usually added to municipal water for sanitation purposes. Carbon will also reduce/remove byproducts of chlorination in addition to many other unwanted contaminants.
If your water source is a private well, without viewing a lab report for your raw well water, we can only speculate on the filter's purpose.
Is there anything indicated somewhere on the tank to hint at its purpose?
Perhaps there is a removable rubber plug on the upper dome of the media tank, a little below the control valve, which will allow media to be more easily added into the tank. This type of tank is most commonly utilized for an acid neutralizer.
How many settings are programmed?
Include some photos showing the front, rear and sides of the control valve, with any covers removed. Also specify the DLFC (drain line flow control) flow rate, usually indicated on a label somewhere on the unit, and indicate the diameter and height of the media tank.