The team, I believe, started in the AAFC in 46 as the Buffalo Bisons then changed names in 47 to the Buffalo Bills. That team played unil 1949 when the AAFC folded and was incorporated/merged into the new NFL. When the city was granted an AFL franchise in 1960, the team kept the name of previous franchise "Buffalo Bills." The franchise was not affliated with the current team.
I also saw the question and, of course, got it right. The final Jeopardy category was "Pro Football" not "NFL Football" or the "NFL." Therefore, the question, as it did, could have included any of the pre- and post-NFL leagues. Dallas Cowboys? That answer (both by women) didn't make any sense.