Sounds like your neighbor has a network using the same SSID as you. BBSIDs are actually the router/ap Mac address. So a router having 2 of them would be unlikely unless it has multiple wireless adapters.
One of the
automatically
You might try setting it up as the Preferred Network in the wireless properties. Once connected stop the windows WZC utility. From a command prompt type services.msc find wzc and stop it. To reconnect you will probably have to enable it again. There are some methods to automated this process, search "wzc stop icon" or so. Or get you a new router that has only one BBSID