As for a mental institution, it is what it is, regardless of the names people think up for it.
Do you refer to a sports institution as an institution, or as a sports club, sports field, pitch, games field?
Do you refer to a health institution as an institution, or as a hospital, clinic, health centre?
Do you refer to a financial institution as an institution, or as a bank, insurance company, pension fund, stock exchange?
I refer to a 'mental institution' as a mental health clinic or psychiatric hospital or psychiatric unit, or, (exceptionally), as a high-security hospital.
I do remember the vast Victorian Asylums and some Convents which were filled with people whom society had strugged to accomodate - including unmarried mothers, sufferers of depression and anxiety, those with learning difficulties and many more who would live their lives there - those were 'institutions' in the sense that I still use the word.