I think the right way to think about this is that saying that someone is "enlightened" is in effect saying they are one specific thing, in contrast to other things--and that would be...just another thing.

If Buddhists texts say "There is no enlightenment. No one is enlightened," the meaning is that enlightenment is not a "specific state" that one seeks to achieve. Enlightenment, really, is nothing other than the ABSENCE of ignorance. Once one ceases being ignorant and deluded his perception can then be referred to as "enlightened." Enlightenment is not a state to be achieved, ignorance and delusion are states to be abandoned.