When it is said that X is an illusion, it is meant that what is perceived as X is really Y. Once the false perception is corrected, we can understand the difference: what we are actually perceiving is different from what we thought we were perceiving.
The correction of the false perception reveals that the perception of X as Y cannot be categorical. Meaning that X cannot always by everyone be perceived as Y, because if everyone always perceives X as Y, then it cannot be known that there is in fact a difference; and thus it cannot be known that X is an illusion. As far as anyone can ever know or say, X is simply X
This is precisely the problem we run into when we posit that consciousness is an illusion. To say that what we perceive as consciousness (the awareness of a singular metaphysical Self) is in reality something else implies that a difference can be known—that the false perception can be corrected. However, since consciousness is a frame of reference for perception, itself, it can never be perceived that consciousness is something else. It can never be perceived that “I” am actually “not I” because the perception of such a distinction requires the frame of reference of “I” in the first place. Put another way, the frame of reference of he who makes the distinction between the illusion and what the illusion actually is cannot itself be the illusion.
A distinction between the “illusion of consciousness” and what consciousness “actually is” is something which no one, anywhere, at any time, can perceive. Thus, any claim regarding such a distinction is necessarily false. Further, I would add that since consciousness necessarily implies choice and will, any claims as to the illusory nature of these things must also be false. Much more could be said of choice and will, of course, but we will leave it for now.