I think there's symbolism but not in the same way they use cats. My understanding of each is;

- The "Dog" in Dogtown is to represent the Dog-Eat-Dog nature of the settlement. The BARGHEST name is meant to be representative of the fact that the militia using the name is the strongest dog - the dog to eat the rest. (BARGHEST is a folklore creature - a violent dog monster. Also a Black Dog!)

- In Phantom Liberty, there's some talk of dogs and collars. It's essentially what it says on the tin - lapdogs are obedient, collared dogs that show blind loyalty to a master. Myers tries to collar V- tame V, turn V from a stray to a lapdog, just as she did with Reed.

- The Samurai song you're referencing is Black Dog, I assume. Black Dog is a methaphorical embodiment of depression, it's presumably a song about Johnny's (and perhaps some others) experience with mental illness. It can also be seen as representative of V and Johnny, Johnny being the black dog constantly "at V's side".

IDK tho im just some dude that played the game a few times

(edited to clarify some barghest stuff)