If the power cord is plugged in but the computer is shutdown, and the light is still on, then that means the network adapter supports WoL or OOB management and must stay on for that reason
Also worth noting that Windows is especially bad about actually shutting down when you tell it to shut down because something something fastboot. I've seen similar inconsistently on Linux but I strongly suspect that to be more edgecases with specific hardware and my install.
So usually one of the lights on the port indicates the link state (up/down and if its at full speed or a reduced speed) and the other light indicates data flow. Both lights blinking suggests either a really shoddy link state or an unusual implementation of status lights on the port. Do both lights blink while its booted and actively transferring a large file? Can you find documentation of how your device implements the indicator LEDs? (I can't tell if that's a dongle or a port on your computer)