Slim doesn't really compare himself to the old dog. What actually happens is that Candy's old dog was suffering and 'ain't no good to himself' as Slim says. Carlson takes the dog out and shoots it for Candy because he can't bring himself to do it. Later, he regrets this because he feels that the dog should have died at his hands out of respect, rather than having someone else do it for him. This sets the stage for what George will have to do for Lennie.
