This kind of talking merry-go-round is the sort of thing we experienced with our now deceased, adult daughter during her long mental illness. We did find it quite difficult for several years to know how to act with she asserted something impossible and stuck to her story. She did become upset when we doubted or questioned some of her statements. If we tried to explore the factual basis for a hard-to-believe assertion, asking when something happened or why she hadn't mentioned such stunning news before, she was often disdainful of such pedestrian questions. Sometimes, she would inform us that the event or the fact was true 'psychically', implying that pitifully under-equipped creatures such as us could not hope to grasp the why and wherefores of higher things such as she dealt with all the time. She did seem happy to have us calmly congratulate her on her up-coming appointment as the head of the CIA and then to go on to discuss the weather.
We continue to be interested in mental illness. It is surprising to me that something like asserting that one is the living Christopher Columbus doesn't sound all that troublesome. The explorer is not a subject of daily comment so what does such an assertion really matter? It matters because first of all, nobody likes a liar. Making emphatic statements that are not true, sometimes statements that COULD Not be true, ruins communication. If someone makes no distinction between what is true and what is not, we would doubt anything they say. We would not have a good basis for friendly social interaction.
But things could go farther in a bad direction. Suppose such a person 'realizes' that I am Hitler or some other tyrant or criminal. Maybe they will realize it is their duty to eliminate me. Our daughter did decide that her boyfriend had been replaced by an identical look-alike. She attacked him for her own safety. She did decide that the telephone wires running along the outside of her apartment building actually led to a bomb and that it was her duty to break them, which she proceeded to do.
Not all mental illness relates to truth and non-truth. It may involve serious, incapacitating depression or undeniable impulses to wash one's hands repeatedly or return home 5 or 10 times to check that the stove is off and the door locked. But it is all frightening and upsetting for all concerned.