@gdfgsdgrrnm2's solution worked for me. Thank you Thank you Thank you so much gdfgsdgrrnm2!
You have to turn off SIP to successfully install the extension, but when it's installed, it's enough to add an exception for kernel extensions from the ASIX manufacturer, so that you can turn SIP on again without breaking it.
reboot to recovery mode (CMD+R at boot) and open up the terminal.
csrutil disable
reboot
install the latest version of the driver from asix (Mac 10.9 to 10.5 v2.18 driver *not the latest 11.1 beta driver*)
allow it in System Preferences -> Security & Privacy
reboot to recovery mode again and open up the terminal
csrutil enable spctl kext-consent add 5RHFAZ9D4P
reboot