Backup up to external hard drive

This sounds like a bus powered (usually rotational) drive that is trying to draw too much power from the USB port. If your drive uses a rotational drive and you are connecting it directly to the computer, one solution would be to connect it to a powered hub - that is a hub that has its own power source.

How is the disk connected? With USB C directly or via a hub/usb a connector? You might want to connect it to usb C directly to see if the issues persist

My M1 iMac started doing the same thing about 2 weeks ago. I was using a Toshiba SSD for my Time Machine backups when I began to get the "Disk not properly ejected" message, and it was the same as your situation, it would occur after a backup had begun. I talked to Apple support last weekend and the agent I was talking to put me on hold while she talked to an engineer. When she came back she said there was a workaround solution, where I reinstalled MacOS after shutting the computer down. Didn't work. I replaced the Toshiba with an OWC Serial ATA SSD without any luck. Both of my SSD's were connected with USB-C. If Apple has a workaround for this then it's apparently a known issue but it doesn't appear there is a solution yet. As of now I haven't run any Time Machine backups in a week and I don't have access to the Toshiba backups which go back a couple of years. Maybe you could call Apple Support to open a case for your problem, you may have better luck than me since we aren't using the same computers or HD's.