You should see something like this as you plug in the USB (while looking at “dmesg --follow”, but this host is actually Fedora, so it may look a bit different):
[12718.226489] cdc_ether 1-11:1.5 usb0: register 'cdc_ether' at usb-0000:01:00.0-11, CDC Ethernet Device, 16:ea:e5:f1:38:5e
[12718.226521] usbcore: registered new interface driver cdc_ether
[12718.250765] cfg80211: Loading compiled-in X.509 certificates for regulatory database
[12718.251349] cfg80211: Loaded X.509 cert 'sforshee: 00b28ddf47aef9cea7'
[12718.275469] cdc_ether 1-11:1.5 enp1s0f0u11i5: renamed from usb0
…in particular, note under cdc_ether “16:ea:e5:f1:38:5e”. This is a MAC address, and output is from a TX2, but what you should see would be nearly identical.
One source of confusion is that the micro-B USB may provide more than one virtual USB device. The “cdc_ether” is key to the specific log line you would be interested in.
Alternatively, you can run nm-connection-editor prior to inserting the cable, and then see if something new shows up. As long as the cdc_ether driver is available nm-connection-editor should see the new entry. Each time you plug in or disconnect you should see a change to the device list of nm-connection-editor.
If this does not happen, then I have to wonder if your installation is working correctly (perhaps the Nano is not running the ethernet over USB).