John, I don't think it is the same tune for both songs, although there may be similarities in the plaintiveness and simplicity and (I think) minor key. I can't check now, though, as the LP of the film version leaves the song out.
It certainly isn't traditional, and I believe it was written specially for the film, in which it is used very effectively, I think. The scene gives me gooseflesh every time I watch it (it's one of my favourite films) and probably did more than any book (of which I've read many, being a history teacher and interested in my country's past) to make me realise just how cleverly and effectively the Nazis used innocuous things to spread their ideology. I can also remember one of my mother's children's books where a family with five children and overworked parents is given a housekeeper who is just too good to be true, gets them all back on the road, is adored by the children and quotes Adolf Hitler every other minute. What a surefire method of indoctrinating (?) a child!
Unfortunately, the song you ask about may be more effective than the authors could have wished: I went for a visit to Berlin a couple of years ago, and one night a guy with a guitar (American, going by his accent) jumped into the underground train I was travelling in and started to sing this song. I was still thinking about whether or not to speak to him when we reached the next station, he finished his song, raised his arm in a Nazi salute, shouted something I can't remember now and jumped out - perfectly timed. Gave me the chills!