I've always known the saying as Happy Holidays especially during Christmas Time.
Because after Christmas Holiday is New Years so saying Happy Holidays would sound better and mean that you wish them happy days for both those holidays.
Also even if you say Happy Holiday on Thanksgiving then the next Holiday is Christmas.
Actually when you wish someone a Happy Holiday there are other holidays coming up.
Either way though works and it doesn't really matter if you say Happy Holiday or Happy Holidays.
It wouldn't matter how someone said it to me but I always said it as Happy Holidays.