The ASUS MB16AC portable monitor accepts either DisplayLink or DisplayPort input. This means you have to get a DisplayPort signal out of the iPhone or the DisplayLink drivers into the iPhone and use the appropriate USB adapter on the iPhone. As I write this I believe there are not any drivers for DisplayLink video adapters or displays.
The Lightning port on iPhones does not provide a video signal like how USB-C ports would normally do, it relies on the adapter to do the heavy lifting. An iPhone with an HDMI or VGA adapter would be told by the adapter to provide a compressed video signal to the adapter, then that adapter decompresses the signal and converts that into HDMI or VGA. It is certainly possible to find a conversion device that will turn the HDMI signal into DisplayPort, but that doesn’t sound like it would be necessarily helpful. A conversion device kind of destroys the portability of the display. It is also possible for someone to build an adapter that can take the video signal that an iPhone produces and turn that into DisplayPort, that would seem actually quite trivial since it has been done with HDMI and VGA, but I have not seen such a product on the market.
Buyer beware with DisplayPort/HDMI adapters as many of them rely on “dual mode” ports to function. These adapters work with DisplayPort outputs going into HDMI mode for backward compatibility, the port is producing the HDMI signal and the cable or adapter is just wires (maybe a signal booster or amplifier) meaning no actual conversion is done outside the source device. I see nothing to indicate the display supports “dual mode” so it is unlikely that it has any kind of HDMI backward compatibility.
A Lightning to USB-C cable will not work in connecting to this display because, again, the iPhone doesn’t produce a video signal all on it’s own. Also, the cable is built in such a way to put the iPhone into USB B-mode (or device mode, as opposed to A-mode or host mode) and the USB-C end of the cable tells what it is connected to it to go into USB A-mode. The Lightning to USB-C cable is a USB A-to-B cable but with different mechanical fittings, it lacks the wires and electronics to pass a video signal. The cable lacks the ability to function for anything but USB and power.
It would be unlikely to find a Lightning to USB-C cable capable of DisplayPort or DisplayLink because it would be too easily confused with a cable for charging and USB data. Apple and the USB-IF group have requirements on cable design and such a cable would almost certainly violate one, the other, or both specifications.
What could happen is a Lightning to DisplayPort cable or adapter and that could be connected to a DisplayPort to USB-C cable or adapter. It seems silly to have to do that from first sight but by digging into the specs it makes a bit more sense. The idea is to maintain user expectations on what a cable looks like and what it does, fail to maintain that consistency and USB-C becomes even more confusing. USB-C is confusing enough as it is.