Some abilities benefit more from one stat than others. There are many pets that have certain attacks that are most efficient paired with a certain stat.
In your Fawn's case, you may choose Power over other stats to increase your Fawn's healing. On the other hand, in the case of a Flayer Youngling, you may choose Speed over other stats because two of its abilities benefit the most from going first (or in other words, having high speed).
In PVE, breed has less impact on the battles. In PVP, breed players a bigger role.
Ultimately it's up to each person to pick whichever breed their most comfortable using in a match. Or not. It's entirely ok to feel that focusing on breeds should not be a priority.