If showering every other day is an improvement then clearly he was showering even less frequently before.

People with no mental illness will smell bad if not washing up. So it doesn’t sound like mental illness is to blame, but even if let’s say it’s, I see no need to build up unpleasant smells.

Daily showers, daily change of clothes, and possibly regular change of sheets/pillow cases might solve your issue. Whatever causes the smell, cleanliness should help to fix it