You can use your chocobo as combat pet in PvE. In other words everyone has an optional combat pet even if they're not a pet class (and the pet classes can essentially have two pets out).
Chocobo vs chocobo fighting definitely isn't in.
Chocobo breeding is being added in patch 2.1 with housing.
Chocobo racing is being added with the golden saucer, but we don't know when that will land.
I think Chocobo fighting is something that's not going to really mesh with the lore. You can fight with them, but they're never going to have "pet battles" with Chocobos. That's not a place they want to go.
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You can use your chocobo as combat pet in PvE. In other words everyone has an optional combat pet even if they're not a pet class (and the pet classes can essentially have two pets out).
Chocobo vs chocobo fighting definitely isn't in.
Chocobo breeding is being added in patch 2.1 with housing.
Chocobo racing is being added with the golden saucer, but we don't know when that will land.