Usually, when there is a "classified" payload, there are hints as to what it would be. Why is it different on this mission? Why is no government entity taking credit for it?
There's really no way to know. There are a number of articles out there speculating, but there isn't a lot really known. What we do know is it was manufactured by Northrup Grumman, will enter Low Earth Orbit with an estimated inclination of 40-60 degrees, and was payed for by an unnammed government agency. There are 2 launches in recent memory that have followed similar conventions. Of these, one of them was later determined to be NSA (I believe I read, can't find the source). It does not belong to the NRO (According to them), who usually launches top secret US satellites. See this article for more information.
If I had to guess, it belongs either to the NSA, CIA, or some not very well known government agency. The purpose could really be anything, from proof of concept to spying to communications, or perhaps more. They probably aren't taking credit because the first thing that anyone will have problems with is identifying who is using it, which makes it even more difficult to figure out what the purpose is. With NRO missions, one knows they are some kind of spy satellites, these are less well known.
update: apparently there will indeed be a Zuma launch:
Some speculate there is no spacecraft for the Zuma mission to begin with.
From Florida Today's article SpaceX launch of secret Zuma mission on hold until after Thanksgiving:
Some have gone as far as to suggest that there really is no Zuma mission — that the pre-launch preparations were an exercise to prove SpaceX's ability to fly on short notice. (emphasis added)
SpaceX put a Falcon 9 on the pad at KSC and test-fired its main engines on Nov. 11. The rocket went vertical again before last week's planned launch attempts, but has since returned to its hangar.
Under that scenario, the payload fairing tests SpaceX cited might merely be an excuse to stand down from a launch that was never really going to happen. (emphasis added)
Time will tell if SpaceX confirms a new launch date for the mission after the holiday.