As commenters have indicated, this use of favor has a long history. It is indeed Middle English (favour) with a French parallel (faveur) - all according to the OED.
The transitive verb meaning
to resemble in face or features; rarely, to resemble generally have the look of...
can be traced to Ben Jonson in 1609, and a parade of others running into the 19th century (thanks and a hat-tip to @RaceYouAnytime).
A possibly earlier noun usage
appearance, aspect, look
is cited as early as 1450 by the OED. Whether or not this relates to "favour" as a ribbon or badge to be worn in competition, it seems to cement this "representative" idea of favor/favour firmly in the history of the language. Interestingly, these definitions are labeled "archaic" or "dialect" by OED. Like many popular expressions, this one must have migrated to the American colonies, where it became entrenched or isolated depending on regional populations.
The rest lies somewhere between history and mystery.