Mental illness is a social phenomenon. There are lots of physical explanations for symptoms similar to psychosis - drugs, infection, immune disorders, chemotherapy, professional head trauma addict, but when nothing else clearly fits, "mentally ill" is the default, the end of the "switch" statement.
The social construction of the category is embodied by the admission process at a hospital's mental ward. They literally go down the various causes that they can check for a disordered mental state, and if they can't find anything else, that is mental illness, end of story. In fact, lots of abnormal blood tests may be ignored if they don't fit into an obvious pattern.
The meaning of the term is defined by what people with the power to assign it do. There is no abstract essence of it beyond the process as enacted.
It's a label that serves a critical function of allowing society to abandon difficult diagnoses and redirect resources to others permanently. It doesn't represent an illness, but the lack of understanding what it is. This is not at all the same as the (unknown) constituent disorders being unphysical or one single entity.
It's sort of like the "scientific creationists" that keep finding God in the gaps of science. There's always something left, but either you have faith in naturalistic explanations or not. If you aren't satisfied with science, you can't talk as though there is such a thing as scientific fact about the scientifically unknown, and implicitly lump it all together.