Irving is a Scottish surname from a river place name meaning "green water," turned American given name, turned occasional pet name. It reached peak human use in the 1920s-1940s, which means it now reads with strong vintage flavor. On a dog, it's a deliberate retro choice: the canine equivalent of naming your child Herbert with full ironic affection.
The Old-Man Name Aesthetic
Irving belongs to the category of intentionally unsexy mid-century American names with a second life as ironic pet names: Earl, Harold, Bernard. It works best on breeds with naturally grave expressions: Basset Hounds, Shar-Peis.
Human Crossover
Irving Berlin, who wrote "White Christmas," is the most culturally prominent bearer. The human name is functionally extinct in current baby naming, which is precisely what makes it available as a pet name.
The Counter-Reading: Requires Explanation
Irving is so thoroughly out of fashion that many people will assume it's a joke name. Owners should decide in advance whether they want to explain the choice or simply let it stand. Most who choose it find the explanation pleasant rather than tedious.