It's a bit too different to translate precisely. The more common phrase you hear in the run-up to New Years is "с наступающим," which means "Happy upcoming (holiday)." I'd say this is more equivalent to our use of "Happy Holidays," since we have in mind the entire holiday season. "C праздником" will mostly be used on the day of the holiday itself, "Happy holiday." Just to note, though, neither of these are New Years / Christmas season specific, you can use them for any upcoming or current holiday.
No, we don't usually have a problem to greet with every individual holiday. Why would we want to compact Hanukkah, Christmas and New Year into one "seasonal greeting"?
I thought that all those "happy holidays" and "seasonal greetings" is a product of "political correctness". If that is really so, then we most probably don't have something similar in our language (and I hope, we never will). We don't do much political correctness here, thank God.
Well, I gave it a thought. And asked my colleagues.
С наступающим is usually used with New Year. That is grammatically correct phrase: С наступающим Новым Годом. Sometimes one can be greeted with "с наступающим праздником".
The only case of multiple holidays that I can think of is "майские праздники" - 1st of May and the Victory Day (9th of May). They are close together and there is a string or two of several non-working days. But we don't usually greet with "с майскими праздниками", V-Day is very special. We just use the term to designate some time period:
На майские праздники я поеду на дачу.
Встретимся после майских праздников.
Also we can use the word праздники to refer to several non-working days connected to one proper holiday. But not in a greeting manner.