How do you delete an old apple TV from showing up?

AirPlay options should show “live” discovered devices. Nothing is remembered from earlier discovery, nothing needs deletion. Could it be that you discovered a device in a neighboring apartment with an accidentally familiar name? If their access options aren’t set to restrict to their own network, then it may show as available to others as well. It may still need a pairing code or password to actually stream to it.

I don’t think there is a method to hide devices that you do not want to see. You can ask for it.