"I've got mental health" can sometimes irritate me a bit, but most people using it are I think just ordinary people using a shorthand term they've come across before, to get over some idea of what they mean without getting too specific about whether they actually have a mental illness or just get nervous/fed up/pissed off from time to time.
What annoys me more is "bipolar" as a noun.
I have bipolar disorder, and I'm fine with expressing it in that way, or by saying that I have manic depression/manic depressive disorder/manic depressive illness/whatever. [Edit: or even bipolar affective disorder, if we want to get properly technical and medical textbook about it, but "affective" can be easily misinterpreted in general communication. "Bipolar disorder" is already shorthand for convenience.] I'm also fine with describing myself as a manic-depressive. And if an adjective is needed, I can say that I'm bipolar, or I'm manic depressive. Some people don't like the older manic-depressive terminology, or take issue with the "a manic-depressive" type of framing (like some people object to "a diabetic" or "a schizophrenic"), and that's fine by me, we all have preferences.
But there's now some weird push to rename the illness as just "bipolar", as in "She has bipolar", or "We run support groups for bipolar". Bipolar what?? It's like saying "He has inflammatory" or "My daughter has an eating".
And if this was just ordinary people using shorthand when chatting among themselves, it wouldn't bother me much. But the actual bloody main UK bipolar disorder charity has now switched wholesale to talking about "having bipolar" in all official communications. Drives me nuts.