From the abstract:
"Authors suggested that infection with cat parasite Toxoplasma could be the reason for this association."
From the conclusion:
"Absence of association between toxoplasmosis and depression and five times stronger association of depression with cat scratching than with cat biting suggests that the pathogen responsible for mood disorders in animals-injured subjects is probably not the protozoon Toxoplasma gondii but another organism; possibly the agent of cat-scratched disease – the bacteria Bartonella henselae."
Not exactly a smoking gun.
There are close to a billion people with cats, and like a few tens of thousands suicides, that include tons of people with no cats at all.
Especially since even your article says "Cat biting and toxoplasmosis had no effect on the depression, and the number of cats at home had a negative effect on depression (p = 0.021)."
So, only cat scratching does? As if the people with many cats at home don't get scratch? I wouldn't place much emphasis on a single study of questionable quality (given ample empirical evidence that there's no big deal).
"Cats cause suicides" border on "no vaccines" territory...