Feeling low or anxious after a bereavement/divorce/loss of job/illness/injury or anything like that is not necessary a mental illness or poor mental health. It's perfectly natural and doesn't need to be tagged with something that suggests something is wrong with them - but that doesn't mean that they won't benefit from support, counselling or therapy.
At the same time, somebody who has a mental health condition (using the phrase 'has a mental illness' sounds unpleasant to me, in part because it can be used to discriminate or denigrate people) can also need support not around the specific diagnosis for similar reasons. There are also personality disorders, challenges around neurodiversity, physical illnesses such as autoimmune disease that intrinsically affect mood without actual poor mental health as a result of uncontrolled inflammation (eg, somebody in an RA flare might have the first sign of it being feeling low and only wake up with their hands like balloons six months later, one steroid injection later and they're fine by the time they get home from the clinic, having been in tears over a hole in a sock that morning), tinnitus/hearing loss (as it can be draining to never, ever have a moment of silence, but they're not depressed, don't need medication, but need some acknowledgement or support).
Sometimes it's about wellbeing, sometimes it's perfectly ordinary mental health but there's a particular thing that needs support to prevent it becoming worse, sometimes it's targeted, sometimes it's poor mental health, sometimes it's something else entirely - getting irritated about an attempt to convey that it's not just for specific conditions without a massive list of circumstances, diagnoses, non-diagnoses and the like seems pretty unhelpful.