I never paired my ATV 3 Remotes so they all will work on any of my ATV 3's. They were in different rooms so never had a problem. They are not that expensive to get a new one. Maybe the remote you have can be unpaired. Maybe one of the ATV 3's needs to be reset (not sure). Also, you could use your iPhone Remote App (if you have an iPhone). I am now all in on ATV 4's where I have to pair all remotes because of the BT.
You must have inadvertently paired one of the remotes, because by default they all work the same. Heck, even the white plastic IR remote that came with my Mac Mini (back in the day) works on ATV3.
It looks like Alrescha provided the unpairing procedure for you. I'd try that.