I believe they come with a usb receiver, which will be synced to the specific mouse. though they do have a way to resync. so you could try to sync the new mouse to the old dongle... but probably easier to just use the new dongle.
As an example I have 2 identical wireless keyboard touchpad things that i use with my HTPCs, and they each only work with their specific usb dongle
Though worth noting that bluetooth is a separate thing, so if you get a bluetooth mouse, you need to check if your PC has bluetooth already. Most laptops from at least the last decade will. But its not guaranteed that a desktop will have it built in.