Damien is the French form of Damian, from the Greek meaning "to tame," which has obvious irony potential as a dog name. The other association is harder to escape: Damien Thorn, the Antichrist in The Omen (1976). Owners who choose this name are generally aware of that reference and find it funny, which says something useful about them.
The Horror Film Hook
The Omen franchise made Damien synonymous with sinister elegance: polite, well-dressed, ancient evil behind the eyes. That transfers to pets with a wicked streak: the cat who knocks things off tables, the dog who stares unblinkingly during dinner. Lucifer and Diablo occupy the same space.
Breed Fit
Damien works on breeds with a striking, intense presence: Doberman Pinschers, Rottweilers, black cats. It falls flat on a golden-hearted goofball.
The Counter-Reading: Hard to Introduce Without a Reaction
Anyone who knows The Omen will react when you introduce your dog. That's either the entire point or a daily inconvenience. The human name has been in consistent use despite the film, which suggests most people can separate the fictional villain from the name's actual warmth.