I think in English you usually say "suffers from depression" when meaning an ongoing illness. Is this what the Greek sentence says? "His wife is depressed" could just mean she is depressed today, rather than a longterm condition. "His wife has depression" doesn't sound quite right in English.
UK native speaker. "His wife has depression" is fine and common. As you say, it implies a clinical / significant rather than a short term condition. "His wife suffers from depression" is also fine and perhaps implies sympathy / regret. "His wife is depressed" is less specific and could imply a short term / less significant problem, but this would perhaps be qualified by "at the moment" or "today".