I think that’s Joshua Graham’s dog.
You are closer than you think OldLady :)
That is Rex, your Cyber-Dog Buddy you get from The King, (he is The King of Vegas' Elvis Impersonators)
You discover that Rex was one time owned by Caesar, the main Antagonist in the Game. Who Founded Caesar's Legion Cooperatively with the Malpais Legate, AKA Joshua Graham.
Rex is a great companion for you, he truly is an Iron Dog :D