It's actually "何{なん}だ", and here 何 works as a stand-in for things that you want to avoid mentioning, or that you think are not worth mentioning, by name. So no, it's not a form of "なのです".
More relevantly, we can think of "XXXだ何{なん}だ" as a set phrase meaning something like "XXX and whatnot".
The "と" right after it is a quotative/complementizer "と", indicating "聖夜{せいや}だなんだ" is what the song repeats (though the substitution of 何{なん} is by our songwriter), and the whole "聖夜{せいや}だなんだと繰返す{くりかえす}" is a relative clause modifying "歌{うた}".
Thus we may translate "聖夜{せいや}だなんだと繰返す{くりかえす}歌{うた}" to "(the) song that keeps on about it being the Holy Night and all that stuff". (The "と" that comes after it is a conjunction: "and")