Is the "az" here because it is a generic statement, or is there another reason ?
"Március egy tavaszi hónap." is correct too. "az" means "that" there and it's often used in places like this. It's like saying "March, that's a spring month." I don't know, it's like "az" is taking over the role of "is" but this is not a scientific observation so take it with a pinch of salt. Point is, you can leave it out, you don't have to worry about it.
We have sth like that in Polish (and I think in Russian too) - Kwiecień to wiosenny miesiąc, to being the "that" part, without any verb or copula. Апрель это весенный месяц in Russian, word for word translation.
This is hard for us in the Southern hemisphere! I have to really think about it.