I think it can be a hassle having to differenatiate between the two Wally's when they're in the same topic or story.
You could try differentiating them by codename but I think more people are liable to use a character's real name then their codename when discussing them.
And I also think it would just be better for the Black Wally if he wasn't constantly called NuWally or Wally II or whatever just to diferentiate him from the OG Wally who's more popular.
Which doesn't make it practical for fiction, in my opinion.
Krypto is the Kent family dog, though he does tend to gravitate to whoever the current Superboy is.