how much impact does Breed make on a wild battle and/or Tamer battle?
Well, it is not nearly as important. On the PvP side, most everyone is going to bring their strongest pets to battle, which are going to be rares. Breed ID's are important as far as being able to out speed the opponent even if as minimally as possible.
When you look at wild pets, if you are battling with a rare pet, it's likely to have stats significantly higher than 90% of the wild pets you face (assuming that they are a similar level). The only time it *MIGHT* matter is if you are trying to farm battle stones with lower level pets.
Tamers are a little bit of a different beast. Breed ID's don't matter when fighting them.... yet. Particularly in Pandaria, with epic and legendary level pets, you have little chance to out-speed them. Breed ID's *MIGHT* be important if you wanted to minimize speed and go for power or health. Otherwise, until we can upgrade to epic pets, trainer pets are going to be faster. TLDR: Unless min/maxing battle pets is something you enjoy, it's largely unnecessary outside of PVP.