I don't think it's equivalent.
If you're taking something like an antidepressant or something, it's because a mental health professional has indicated that they think you'll be better-able to deal with your issues with this thoroughly-tested, regulated item. At least in theory, the professional will be much better-suited to find the exact medication that will help you emotionally regulate.
I've never done weed, but I have gotten drunk plenty of times when I was young (though not in quite awhile), especially when I got dumped by a girl that I really liked. Alcohol can be great in a very temporary sense to make you feel better and ignore your problems, but it's terrible at actually addressing the core issues, and it has a ton of other awful side-effects that might even make the underlying problems worse.
I already said that I have a hatred of weed culture, and I stand by that, but as someone who does take mental health meds for depression, I do understand how appealing coping mechanisms can be, and I absolutely sympathize, but I think there are enough differences between prescribed meds and recreational drugs to where I don't think it's fair to claim that the former is a coping mechanism.