Yes. Happy holidays is used specifically around December to refer to the various holidays taking place then and to avoid excluding anyone's traditions. Other holidays are usually referred to by their name (happy Easter, happy Mother's Day, etc). In Russian though, it's quite common to just wish someone happy holiday without naming the holiday because it's assumed you would already know what day it is.
Have a good/happy holiday! would probably be the more native translation when the specific name of the holiday is not given. The Happy+X construction feels like a relatively closed set of expressions rather than a generalizable formula, at least in American English. I still cringe a bit when I hear Europeans saying "Happy weekend!"