You and your neighbor may have to coordinate this:
First, remove the batteries from all of your transmitters, then unplug the opener unit. Wait 5 minutes. This should clear out all the codes between the opener and transmitter. Then, put in the batteries, plug in your opener, and program as directed, making sure your neighbor stays away from his garage or car.
If that does not work, then your neighbor will probably need to do the same with his setup. Unplug and pull the batteries from yours and his (as stated above), then program his first so as to keep peace in the neighborhood.
If for some off-the-wall reason none of this works, then I would return your genie transmitter and replace with the proper Sears unit.