It is likely legit, and yes, they can be that quick. They were even quicker during the early action application process. In part it depends on the interviewer, some are busy and don't get to the applicants as soon as they are supposed to, others contact their interviewees immediately. Harvard uses alumni to do interviews, and most do not use a Harvard email, they use their business or personal email when contacting applicants.
In most areas, you are sent a questionnaire that asks questions about your grades, SAT and other scores, and a couple of questions. (Sometimes the questionnaire comes from the alumni in charge of interviews in your area, but as far as I know, not from Harvard directly.) You bring the questionnaire with you to the interview or some may have you return it by email. They use the questionnaire to help them have something to speak with you about.
It is unlikely that your friend would get the same email as Harvard alumni typically only interview a few applicants each year and typically not more than one at the same school, at least that's how it works in my area. Be sure to contact the interviewer quickly, as you may have to go back and forth a couple of times to set a date and time (if you're as busy as most Harvard applicants are), and the interviewer is under pressure to submit their report asap.
I really wouldn't bring up that you were concerned that they weren't legit, considering all the facts and that it does say on the Harvard website that you may be invited to interview with an alumni. It seems a bit paranoid and it may make them feel uncomfortable. Being invited doesn't mean that you are higher up in consideration than anyone else, it just depends on if they have an alumni available. I don't know if Harvard knows who is interviewing, as typically all of that is handled locally.
(I know quite a few Harvard alumni who interview applicants.)
Good luck!