I think it's called a "counter-duet" because they're singing TO each other using different tunes for each part.

Not sure if there is anything traditional in the form, but I know of a Gaelic song which has two parts, like this a male and a female part, but the same tune is used for both halves. The song is An Téid Thu Leam, a Mhàiri or in English, Will You Come with Me, Mary. The man is asking the woman to go with him to his home as his wife, and telling all about his land. She's saying she would miss her family, etc. Finally, in the last verse, hers, she says, STOP, I'll go with you.... (Hopefully that implies a happily ever after). The longer song isn't sung much. The Rankin Family on their Grey Dusk of Eve EP sang it as a solo female song, alternating the male and female parts with the one performer, and the whole group singing the chorus.