UPDATE: As of 5/21 there is a driver available at asix.com.tw that is now compatible with MacOS Big Sur on both Intel and M1 macs. With this driver installed, the device functions perfectly.
The chip at the heart of this dongle is the Asix AX88179. Up until the release of Big Sur, this was not a problem. But now Asix does not have a driver that works properly under Big Sur. They do have a driver you can download, but part of the installation calls for you booting into the recovery partition and DISABLING SYSTEM INTEGRITY PROTECTION. That is like buying a new car and using a crowbar to remove all of the seatbelts and airbags. It's a terrible, terrible thing to ask customers to do.
What's particularly infuriating is that Asix had all summer to write new drivers under the new USB ethernet APIs that Apple released (specifically so that they could lock down ring 0 kernel extensions as a security improvement). To some extent I do feel their pain - Apple has a bad track record of moving the API cheese a lot. But even so, as I said, they had all summer long to come up with a proper solution and still, even with Big Sur having been out more than a month.
Oh, and don't expect the driver they do have to work with the new Apple Silicon macs either.