It's a borderline insane branding of this corporate tool. Words and stories apparently mean nothing to these people, so if allowed they'll probably destroy the lot of it for all of us.
Garak is an interesting and beloved character in the series because he is complex, problematic and express it with a convincing sophistication. The gay innuendos help too. He does nasty, deceitful things. He starts a war because it's too grim and disgusting for his close neighbours to go through with it, and it's expected to possibly help fend off a godlike existential threat to the entire quarter of the galaxy.
He's a monster in a suit, a Franz Stangl. I think it's a very, very weird character to associate a corporation with.