Depends on who you ask, for me, they count as both climbing and training.
I have a MoonBoard at my gym, and I usually prefer to climb on it rather than training on a hangboard. Though I do have at least 1 session a week for hangboarding and campus runs, I do prefer to climb on the MoonBoard because it's just more enjoyable compared to just hanging off of an edge. On a MoonBoard, you are also training crimps, and pinches, and you are also just climbing. Not to mention that you also train your core and back and other muscles while doing it, in comparison to hanging on a hangboard, where you train mostly your fingers, and just hang.
So for me, it is training, and it is also climbing, even though the climbing on it is very different from climbing on a regular climbing wall, but every so often you get a boulder route that is very similar to a MoonBoard problem, and if you climb regularly on a MoonBoard or other board wall, those routes will be much easier.
