I also have this Type-C Multi Port Adapter, and cannot get the cable internet connection to work either. I'm running a 2018 MacBook Pro with OS 10.13.6. I enabled the extensions, but for the AX88179_178A, on the System Information page it is not showing as 'loaded':
AX88179_178A:
Version: 1.12.0
Last Modified: 1/6/19, 7:26 PM
Bundle ID: com.asix.driver.ax88179-178a
Loaded: No
Get Info String: ASIX AX88179/178A USB3.0/2.0 Gigabit Ethernet driver
Obtained from: Identified Developer
Kind: Intel
Architectures: x86_64
64-Bit (Intel): Yes
Location: /Library/Extensions/AX88179_178A.kext
Kext Version: 1.12.0
Loadable: Yes
Dependencies: Satisfied
Signed by: Developer ID Application: ASIX ELECTRONICS CORPORATION (5RHFAZ9D4P), Developer ID Certification Authority, Apple Root CA
Help?