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It sounds like you’re talking about confirmation bias. This is when someone seeks out — or in this case interprets —evidence in a way that “confirms” for them what they already believe to be true. I would avoid ascribing this to pathology (such as dementia or psychosis) as these are clinical terms that refer to specific states or disorders, whereas all humans are subject to confirmation bias; it can certainly be annoying when someone behaves this way but usually it is not pathological. Cognitive biases which act as development or maintenance factors for a disorder are usually referred to as cognitive distortions.
I would suggest maybe adding an example next time as the scenario you’ve posed is a bit convoluted. Hope this helps!
Strictly speaking by the definition of cognitive bias, yes. They make their own subjective reality without reason. "I think you are X, therefore you are X" isn't rational because there is no evidence that suggests that what you think is the objective reality. The person could be X, or Y, or Z, or U, there is nothing said about why, just that you think that its whats happening. You see this in real life with political and religious dogma.
As for whether it counts as a psychoses or some sort of illness, not at least to the current criteria in the DSM-5 and ICD-11. A delusion is a false belief that cannot be shaken no matter how much evidence you put in front of them. It is usually present throughout the persons psychotic episodes and is a thought disorder.
While this can be present in a delusion "I think you are a poisoner who is sent here to kill me, therefore you are a poisoner sent here to kill me" it's not by itself psychoses. Each case must be carefully analyzed by a psychiatrist to see whether it is just a case of the cognitive biases that engage us in daily life, or if it's a legitimate concern.