Mental health problems being caused or exacerbated by, or associated with, alcohol use is way more complex and multifactorial than a simple word can cover, though.
When doctors or scientists say that alcohol is a depressant, all they mean is that it's a central nervous system depressant, along with opiates, benzos, and some other drugs. Drugs in this group suppress brain activity leading to slowed breathing and heart rate, meaning you can pass out, go into a coma and even die if you take too much, or if you take more than one CNS depressant at the same time.
The mechanisms by which alcohol can alter your mood in the short term or affect your mental health in the long term aren't so easy to sum up. Some people get giggly and silly when they drink, some morose, some violent, some relaxed, some chatty, and often even for an individual the effect depends on how much they have, who they're with, how they felt beforehand, and lots of other factors. The long-term effects of repeated overuse of alcohol aren't as simple as just depression either. People respond differently and have different problems from it, and some will get depressed just as you can get depression as a comorbidity to many other disorders or as a result of feeling like crap or trying to modify how you feel with drugs. It's not that useful to assign alcohol a word that means it's a drug that will give you the mental illness of depression.
But everybody will, if you give them enough alcohol, experience CNS depression, so it's reasonable to describe it as a (CNS) depressant.