Broadly, yes, dongles and Mice come as a matched unique pair and they can’t be mixed. This allows a computer to support more than one mouse, not unusual if the user is into CAD or video editing etc, and also allows a few hundred computers in a open plan office to work without confusion.
The wireless link is in the same band, 2.4GHz, in which many channels are available. But as many devices are allowed to share the same channel, pairing doesn’t depend only on frequency. Each mouse/dongle has a 64bit identifier that is picked out by the computer; it only responds to matching mouse/dongle pairs.
It is possible for a manufacturer to generalise the match process by providing software allowing a special dongle to be configured by the user. But as far as I know, the facility is proprietary, and doesn’t even work on everything sold by the same vendor.
I know it’s terrible hard on a thrifty model engineer, but the answer is to buy a new mouse. A few months after it arrives the gremlins will rub salt in the wound by unveiling the missing one. It will be hidden in plain sight just were you left it…
Dave