I have a wireless backup system on my home network. in the basement, I have an old Mac Pro, with several big drives on it, always-on. (Has to be running 10.13 or later.) That older Mac has been set up as a Time machine backup destination of the various Macs in the household.
So My wife does not have any cables going to her computer except Power, and her computer does Time machine backups to two different drives, not even on her computer.
you could set up a similar system using a Network Attached Storage NAS) device, or an old Time capsule. You MIGHT be able to use one drive on a USB cable on a router, but I think each Router would be a distinct case for setup. Non-apple solutions do not need a Mac-formatted drive, but they MUST be capable of creating Mac sparse-Bundle disk image files.