It works without problems between different Windows computers (no matter whether purely Bluetooth, purely USB or a mixture of one computer Bluetooth, one USB). There are simply two assignments for the mouse.
Unfortunately, I can't tell you anything about Ubuntu. The best thing to do is to ask Logitec directly...there are certainly one or two people who also use Ubuntu.

Ubuntu is no problem. I use it every day with this mouse, with the USB adapter. And BTW Android also works without problems, via ordinary Bluetooth. The switch also works without any problems. The same applies to the MX Keys keyboard from the same manufacturer.

The USB adapter is an ordinary USB 3.0 type A. However, the mouse can be reliably connected via Bluetooth.