Is there any FAR defining an "introductory" or "familiarization" flight?
There's no formal definition or regulation that I could find; introductory flights aren't defined in 14 CFR 1.1, for example. I'm not sure why the FAA would care how often you fly with someone, or what the difference is between an introductory flight and any other flight with a non-paying, 'non-logging' passenger.

The only other people who might care about it are the TSA but if you aren't providing instruction during the flight then that shouldn't matter either. Introductory flights are usually understood to be a "demonstration flight for marketing purposes" (49 CFR 1552.1) and therefore exempt from TSA requirements, but there doesn't seem to be any rule about how many demo flights you can make with an individual.