Yes, addiction in general is considered poor mental health, even if it is a self imposed habitual disposition.
It is one of the most benign things you can get addicted to though, as it's generally a safe environment. The only major risk is inactivity, with a secondary financial risk; I also assume everyone here understands this.
I'd disagree, it isn't just inactivity, in extreme cases people have died after long gaming sessions. Its like saying gambling addiction isn't too bad as you're only losing money, you still have your health etc.. when it can ruin relationships, push people to do things they'd not otherwise do. I'd suspect that serious gaming addicts can have similar impacts on their relationships, career etc.. and indeed put themselves at risk of further mental health issues.
Ref another poster asking if you can be signed off work for it - I think you're more likely to be sacked for it. Just because it is an addiction doesn't mean your employer needs to tolerate it, there are aspects of personal responsibility/choice too that lead to such a situation. Likewise if you become an alcoholic and turn up to the office hammered or are addicted to drugs then I think plenty of employers will simply sack you if it leads to issues at work. On there other hand secondary issues that might stem from that, depression etc.. might well be a reason to be signed off. But I'd wager that if you were to call in on a Monday and say "sorry boss, I relapsed, spend all night playing RDR2, can't make it in today" then you might as well have said "sorry boss, went clubbing all weekend, on a massive comedown now, why do I keep dabbing when I take MDMA... FML" then they'd be equally unimpressed.