.
On their own they generally do not or cannot realize it. It usually takes an external influence--one the sufferer trusts--to point out that certain facets of the person's psyche "aren't right" or aren't consistent with what is known about the individual thus afflicted. "Brainwashing" works by invalidating sequential or temporal cognition (memories and any/all thoughts relating thereto) and generating replacement delusions which inoculate themselves against reproach by reality, significant others, contrary evidence, etc. Deprogramming a truly brainwashed individual can take years of intensive psychotherapy.
I have a close friend and neighbour who fell in love with a man 30 years ago.
He founded a cult based on the ideas of Adi Das Sam Raj and formed an offshoot group in Australia, which moved into my local area.
It took quite a while after meeting my friend to realise that she was in a cult and deeply brainwashed.
The crunch came several years later, when she was unable to keep up with the mortgage on the mortgage on the mortgage for the property that protected a portion of the cult's interests. She was forced by the bank to default and ended up being expelled by the cult.
She took a social working course, and during the discussion of the characteristics of a cult suddenly work up to the reality of her life - an incredible shock that she is still reeling and recovering from.
Another cult was infiltrated for three years by journalists in England. The book they subsequently wrote was so easily recognisable as true that the cult fell apart from members defecting.
So there are indeed at least two ways for the brainwashed to realise it - and probably many more.