Some controllers do indeed work on multiple voltages, but in your case, the implied 36v/48v compatibility might just be a bad translation of Chinese to English. It could be intended to work on a specific voltage.
To eliminate the need for multiple iterations on these controllers, there might be a jumper on the circuit board that sets the working voltage.
I have one (non-KT) that was available in either 36V or 48V, versions, but it came with an LCD so I assumed I'd could buy the 48V model and run both. Well, I was wrong. I initially set it up on 36V and wasted several hours going thru all 36 Hall combos but the motor just stuttered. LCD worked though, and I could set LVC to 30V. Then I tried 48V and it spun up my motor as advertised. Later, I opened it up and spotted a bare pad marked "36V". When I jumped it to ground, I had both 36V and 48V operation. Maybe all it does is override a hard wired LVC, and the LVC on the LCD is meaningless.
I've found that sometimes there is enough residual charge in my KT controllers that they get confused when I change from a depleted 48V battery to a fresh charged 36V battery. They power up with a low battery icon. Unplug. Wait. Reconnect. Then they sense the proper voltage.