As long as one is not supporting an false beliefs by wishing a happy new year, it’s not haram. When a believer wishes a happy new year, essentially he is praying for others. He is saying: may Allah bless you with a new year. That’s the basis of his intention, so it is permissible.
A bid’ah is when we add something to religion or make false claims. There are many things the Prophet (s) didn’t do that we do today. We drive cars and fly planes. We use phones. And so on. Does this mean they are Bid’as? Of course not. As long as we are not supporting any pagan or false beliefs, and we are not making false claims about Allah and religion, then it’s permissible