Are USB dongles bidirectional?
I have a Logitech mouse that I can map the buttons with an app when in Light Speed mode, but my wireless Akko keyboard requires the cable to be plugged in to be able to remap the keys. Since the dongle is called "receiver", I kinda assume it can only receive signals but cannot send them, but that doesn't seem to be the case with the Logitech dongles.
This can really go either way and it's up to the manufacturer. Also depends a lot on how the device is designed.
The Logitech remapping of buttons is generally done on software side, that is, the mouse sends the same button no matter what and it's the software that interprets it when it reaches the computer to send a different input to the software. Sounds like your keyboard is doing it in hardware, that is, when you change keymaps the keyboard actually sends a different signal to the computer rather than having a piece of software intercept the signal and send a different one to the operating system.
If I were to guess, I'd say the manufacturer of your keyboard probably put a programmable controller, but the wireless function is a basic off-the-shelf wireless keyboard chip and dongle that they purchase off the shelf from a supplier rather than design themselves. The USB cable lets them reprogram the keyboards controller, but the off-the-shelf wireless keyboard chip and dongle don't have a reverse channel to send programming to the keyboard. For a situation like that I would question why they don't use Bluetooth instead of a proprietary wireless system as that would give them an easy programming channel.