There is no concrete answer to this question as the characters' backstories have never been fully fleshed out by the show's writers. However, many fans believe that Damon and Jesse are not brothers, but simply two friends who grew up together in America. This theory is supported by the fact that the two men have very different last names (Salvatore and Brooks, respectively). Additionally, Damon has never referred to Jesse as his brother on-screen, instead calling him his friend or buddy.