There's a difference between a complete, grammatically-correct sentence and a greeting.
If I was writing a complete sentence, I would write "I wish you a merry Christmas and a happy New Year." But if I was just greeting someone, I'd say, "Merry Christmas!", not "A Merry Christmas".
It's like when you write a title or a headline, you often leave out words, especially articles and versions of the verb "to be". A newspaper headline will say something like "Stock Market Up", not "The Stock Market is Up".
I suspect people write "Merry Christmas and a Happy New Year", i.e. no article on the first but including an article on the second, because they're starting out writing a headline type of construction but when they get to the middle it's getting kind of long and they put in the article as if it was a complete sentence through force of habit. I think it's inconsistent and I wouldn't do it. I'd generally write "Merry Christmas and Happy New Year!", or maybe possibly "A Merry Christmas and a Happy New Year."
Of course, in these politically-correct times, the correct greeting is "Happy Unspecified Holiday to you, unless the very idea of holidays offends you, in which case please pretend I said nothing and walked past you in silence".