AFAIK, It should work since Android kernel config select CONFIG_USB_RTL8152 which is compatible with RTL8153 chipset. I already used a Trendnet TU2-ET100 which is compatible with generic CDC Ethernet driver.
What happens when you plug your device to the Hikey960, any message in dmesg ?
Note: Remember that Hikey960 is able to operate in only one role at a time, host or device. You need to have the USB type C (e.g. adb debug cable) disconnected to act has a device.