"Mental health" is a really vague and dangerous concept. We know from reason that the mind is a separate entity from the brain or any other piece of matter and thus it cannot be "handicapped" or "sick" in the way the body can, so we should be very suspicious from the get-go whenever someone uses this term.
Often people point to alleged "mental illnesses" to excuse someone's bad behavior, saying it's not their fault because they just can't tell right from wrong. Such a claim - that some people are simply incapable of being good Babies don't lack a conscience; they lack understanding that other people have feelings like they do. It is impossible to not pick up on this after a few years of living among others. - is all of an affront to the basic assumptions of morality, the worst possible way of dealing with people who behave badly by saying that they can't be corrected and also can't be held responsible, and incredibly dangerous in terms of its effect on society (see below).
The same concept is also, even more sinisterly, quite unironically used to advocate preemptively taking away people's liberty without them comitting any crime. In less extreme examples, it functions as a way of discrediting people with unpopular opinions without dealing with their arguments. There is no rebuttal easier to give, less valid or more insulting than "it's not your fault you think that; your reasoning unit is broken". And yes, I've been called mentally disabled before - by my family - and, in a beautiful irony, it was for making this very claim.
People also sometimes use the term to mean things like depression. That's still messed up and wrong. Depression is an emotional problem with emotional causes that needs to be solved through emotional means, not medical means. And usually the only person who can do so is ultimately the depressed person themselves. You're never going to feel better if you keep sulking and doing nothing of value and expecting a pill to change your mind. You have free will. YOU have to change your mind.
Yet another way the concept is harmful is how it can easily be used to justify censorship. For example, I've heard people defend legislation prohibiting "hate speech" on bases like, "calling me the wrong pronouns is an attack on my mental health". It's crazy, but revealing. If you say that people's emotions are a matter of health, then it becomes difficult not to justify this, because no one can say where "mental health" ends and just hurting someone's feelings begins, because the concept never existed in the first place.
I might use this phrase or "psychological health" occasionally to describe something that simply threatens a large degree of long-term emotional suffering (like the importance of agency), but I mean nothing special by it and I use this language only when I can't find any other phrase that does the job. I try to make sure I'm using it in ways that don't risk insinuating any of the above ideas.
Reading over this I noticed it probably sounds like I just backpedaled on what I said about depression. I didn't. When you use the mental health language to talk about the cause of an emotional problem, and you make it clear that cause is emotional, you're not insinuating dangerous ideas; but when you use the language to talk about the sadness itself you're lending to the idea that this should be thought of in a similar way to real health: caused by something outside the person's control and should be fixed by a physical item rather than a change in behavior or situation.