Connecting a MacBook Air to an ASUS MB166B Portable Monitor

I clicked a link on the site to show all downloads, and now it is showing four driver packages dated 2022/07/07.

• USB Monitor Driver 5.2.1 for Mojave and Catalina
• USB Monitor Driver 4.3 for High Sierra and below
• USB Monitor APP 1.5 for Catalina, Big Sur, and Monterey
• USB Monitor APP 1.2 for Catalina and Big Sur

I think you want to try the highlighted one – the titles suggest that perhaps they rewrote their software so that it runs as an application (user space), rather than as a driver (system space). The fact that they only mention Big Sur and Monterey in the context of the APPs also suggests that the APPs are newer (despite lower version numbers).

I have no idea whether the 1.5 APP will work on Sonoma, but of everything listed, it looks like the most likely to work.

https://www.asus.com/us/displays-desktops/monitors/zenscreen/zenscreen-mb166b/

Apparently this device does not have any standard video input ports. It takes input over USB 3.2, and so there has to be a corresponding driver on the computer (similar to how DisplayLink devices work).

The description of the driver says

MB Series USB Monitor Driver for Mac OS 10.14 Mojave & Mac OS 10.15 Catalina

This driver enables you to use ASUS MB series monitor with Mac OS system via USB 3.0 signal.

If your operation system is upgraded to Mac OS 10.14 Mojave & Mac OS 10.15 Catalina, please upgrade and install this driver accordingly. If your operation system is still staying at Mac OS 10.13 or before, please keep using driver 4.3 version.

I wonder if (a) you have an Apple Silicon Mac, and (b) the driver package is installing an Intel-only kernel extension. Apple Silicon Macs cannot run Intel-only kernel extensions. (That's one of the limitations of Rosetta 2 – it can only deal with application code that runs in user space, not with system code that runs as a privileged extension to the operating system.)