What type of controller is this?
I don't know of any particular name for this kind of controller.

The advantages of a fixed-slope system are great simplicity and inherent stability in most cases.  As long as the output has settled a good way to the new steady steady state each loop iteration, the system will be stable and it won't overshoot.  The drawback is that response is slow compared to more sophisticated control methods using the same plant.

The hysteresis prevents the system from "hunting" around the setpoint and avoids constant small tweaks.

This kind of controller is useful when the thing being controlled and the input signal are largely intended to be fixed.  The controller is then compensating for variations due to temperature, wear, and the like, not actively trying to put a specific "signal" on the output.

Note that this system is NOT a integrator.  A real integrator adds a fraction of the offset (desired - actual) to the control signal each iteration.  This system adds a constant regardless of how far off the output is from the desired level.