Heya HarukoDK ^^
When you respec, you won't lose your rank to 0. You have worked hard to get your companion to Rank 10. It will just reset the SP (Skill Points)
And yes you might be correct with breeding. But still, every chocobo rank 10 on a single line will unlock their barding, unique to them. You will not be able to use the barding of a black chocobo that you made attacker on your first chocobo that you decided to make a Healer. (If it was, would be awesome ^^ )
We'll see and be fixed soon! *Excited